There seems to be two views of the church, one ancient and one modern. The modern came into the forefront at Vatican II, but it was being taught before then by theologians.

The ancient view is this: the Catholic Church is the visible sign of salvation, the visible sign which tells us who is in the way of salvation and who isn’t. Only those within the Church can truly know, love, serve, and live in a manner pleasing to God. Belonging to the faith and unity of the Church is absolutely necessary for salvation. Separation from the visible Church is separation from Christ.

The modern view is this: the Catholic Church is the most perfect guide to eternal life, one must belong to the Church to have the best chances of being saved, and of living in a manner most pleasing to God. However, it is possible to know, love, serve, and please God outside of the Catholic Church, though it is much more difficult do so without the teaching, the liturgy, and the sacraments of the Church. One can be visibly separated from the Church yet united to Christ invisibly through an implicit or explicit desire.

On the side of the ancient view you have all the ancient fathers, popes, doctors, magisterial statements, which clearly express that one must be Catholic to be saved. Some theologians teach that one can belong to the Church by desire, but that desire must be explicit and one must at least believe in Incarnation and the Trinity.

On the side of the modern view is the modern theologians, beginning a few centuries ago and particularly with Jesuit theologians, the most recent Council, recent catechism, recent magisterium.

One way to solve this problem is to say that the modern Church is heretical and anathema, and to avoid it, to separate from the visible Roman hierarchy which is said to have lost the faith by allowing this modern view to be taught.
Another is to say that the modern view has not been proclaimed with any (infallible) authority by the magisterium, and that one must resist this modern opinion that has spread among the members of the Church while remaining loyal to the Church’s hierarchical authority.
Another is to say that the modern view is actually the same as the ancient view, only updated and more refined by the progress made by theologians, and that one must in any case be obedient to the living authority of the magisterium, which has now taught the modern view.

I think that the distinction between the two views is best made as follows: the ancient view states that membership in the Church is a necessity of means, the modern view states that membership in the Church is a necessity of precept. A necessity of means is one where you absolutely have to belong to the Church in order to be saved, because the Church is the means of salvation; a necessity of precept is one where membership in the Church is practically necessary for salvation, because it is hard to be saved without the Church, but not absolutely so.

Note that the proponents of the modern view are forced to say that membership in the Church is a necessity of means, because this is a thrice-defined dogma (Extra Ecclesiam Nulla Salus, Outside the Church there is No Salvation). However, they equivocate on this expression and twist it so that it practically means that membership in the Church is only a necessity of precept, because one can be a member of the Church by an explicit, or even an implicit, desire, i.e. if you are truly contrite for your sins, but have never heard of the Catholic Church, then you belong to the Catholic Church implicitly by your being perfectly contrite for your sins. This practically means that one does not have to be a Catholic to be saved, and that being a Catholic is only a precept, it is “advice” and not a commandment. The notion of “membership” in the Church has gone from meaning those who profess the same faith, participate in the same sacraments, and are subject to the same pontifical head, to those who have good-will in their souls and are therefore implicitly or explicitly are members of the same Church. It is not necessary to belong to the “body” of the Church in order to belong to the “soul” of the Church, in the modern view; whereas in the ancient view, the soul is the form of the body and does not wander about outside it as an invisible ghost.

Note also that it’s not the case that all “traditionalists” hold to the ancient view and reject the modern view. Archbishop Lefebvre was a "modernist" in this sense:

Does that mean that no Protestant, no Muslim, no Buddhist or animist will be saved?  No, it would be a second error to think that.  Those who cry for intolerance in interpreting St. Cyprian’s formula, “Outside the Church there is no salvation,” also reject the Creed, “I confess one baptism for the remission of sins,” and are insufficiently instructed as to what baptism is.  There are three ways of receiving it: the baptism of water; the baptism of blood (that of the martyrs who confessed the faith while still catechumens) and baptism of desire.

Baptism of desire can be explicit.  Many times in Africa I heard one of our catechumens say to me, “Father, baptize me straightaway because if I die before you come again, I shall go to hell.”  I told him, “No, if you have no mortal sin on your conscience and if you desire baptism, then you already have the grace in you.”

The doctrine of the Church also recognizes implicit-baptism of desire.  This consists in doing the will of God.  God knows all men and He knows that amongst Protestants, Muslims, Buddhists and in the whole of humanity there are men of good will.  They receive the grace of baptism without knowing it, but in an effective way.  In this way they become part of the Church.

The error consists in thinking that they are saved by their religion.  They are saved in their religion but not by it.  There is no Buddhist church in heaven, no Protestant church.  This is perhaps hard to accept, but it is the truth.  I did not found the Church, but rather Our Lord the Son of God.  As priests we must state the truth.

The ancient view appeals to those who believe in the mysterious nature of predestination as taught by St. Augustine; to those who want to preserve doctrinal clarity; to those who greatly revere the authority of the fathers, the doctors, the popes; to those who see a continuity in tradition and immutability of dogma as essential to the faith; to those who see the Church in an Incarnational manner, as the Body of Christ, as the visible means of salvation.

The modern view appeals to those who think the idea that only Catholics are saved is proud or bigoted; to those who cannot stomach the notion that so many are lost; to those who wish to be on friendly, easy-going terms with people from other religions; to those who see dogma as something that needs to be clarified by expert theologians through time; to those who think God’s power or God’s love is somehow limited if God grants salvation only to Catholics; to those who see the Church in a “spiritual” or “mystical” manner, operating invisibly all over the world.

The ancient view is called by its opponents Jansenist or sometimes Feeneyite. The modern view is called Pelagian or Modernist.

One objection to the ancient view is that is the cause of despair because it means so many are lost.
One objection to the modern view is that it leads to religious indifferentism, because if it is possible to be saved in a non-Catholic religion then the call to convert to the Catholic religion is not as pressing of a necessity.

It seems that this ecclesiological matter is the most important debate in the Church today. We are desperately in need of doctrinal statements to clarify this ambiguity in the Church's ecclesiology, and a Summa of Catholic ecclesiology which will once and for all sum up what is the Holy Catholic Church.

The Roman Catholic Church has inner and outer bonds of unity.  The outer bonds of unity are what the members have, profess the Faith, partake of the Sacraments and subject (willing to be if none are apparent) to legitimate ecclesiastical authorities.  

The inner bonds of unity are the gifts and fruits of the Holy Ghost, the three theological virtues (sanctifying Grace).  This is the most important bond as this bond determines whether one who dies in this state joins the Church suffering or Triumphant at death.  Some members of the Church do not partake of the inner bonds of unity and are damned.  Some non-members do partake of the inner bonds of unity and are saved within the Church.

Yes, but is it impossible to possess the inner bonds of unity without possessing the outer bonds of unity, as the ancient ecclesiology states, or is it possible to possess the inner bonds of unity without the outer bonds of unity, as the modern ecclesiology states?

Both agree that it is possible to possess the outer bonds and not the inner bonds, i.e. to be a Catholic in a state of mortal sin.

Since the beginning BOD and EENS has been taught.  The only possible way this can be true is if non-members can be saved within the Church.  This could be very rare as I am not sure how common it is for a non-member to die with a supernatural Faith and perfect charity but it is possible.

What confused the issue even before the false ecuмenism and universal
salvationism of V2 was a mis-application of the terms "body" and "soul" to mean one could be a member of a Church which was wider than the Mystical Body of Christ which is one and the same as the Roman Catholic Church.  The "inner bonds of unity" is not the "soul" of the Church.  There is not a mystical Church that is wider than the Roman Catholic Church.  There are not "members" of the mystical body of Christ who are not members of the Roman Catholic Church.  There are non-members of the Roman Catholic Church who are within the Roman Catholic Church by desire.  This is what has always been taught.  This is what Catholics are obliged to accept as God does not unjustly condemn or save anyone.  No one is damned to the pains of Hell without it being their own fault and no one is saved who dies without being in a state of sanctifying grace.  We should be able to accept that and move on to the task of saving souls as anything beyond this is obstinate wrangling.  

Here, I will quote from article which demonstrates the modern ecclesiology well, an article on Pope Francis' idea of "ecuмenism of blood", that heretic and schismatic "Christians" who die for their "faith" are martyrs, every bit as much as Catholic martyrs are martyrs:

During his address to the movement members, Pope Francis referred to the suffering of those who shed their blood for Christ, pointing out that, although they may not be Catholic, they are our brothers and sisters because they died for Christ.

[. . .]

If I’d read the Pope’s words earlier in my life, I likely would have been scandalized, not out of animosity, but rather out of naivete.
There’s a grave misconception that Catholics believe that they are the only ones who are “right” and that anyone who’s not Catholic goes to hell. Some Catholics have believed that in the past, and some still do believe it.
What the Church actually believes is that the Catholic Church has the “fullness of Truth.” All other Christian faiths have some of the Truth to various degrees, and we must recognize that.

[. . .]

My life experiences of getting to know Christians of other denominations has helped me to better understand what is meant by ecuмenism.
It really came to life for me on my recent trip to Jordan as part of the Jordan Tourism Board Religious Journalist/Blogger Tour 2015. We were indeed an ecuмenical group – all Christian, but not all Catholic – and I loved it. It gave me the chance to see Christ through their eyes, learn about their faiths, and share what we had in common.
I was quite surprised – we had an awful lot in common!
I also had the chance to see the refugee camps with the rows upon rows of simple shelters built for the refugees by the Jordanian government and to meet with a few of them being treated at the Italian Hospital in Amman.
When you look these refugees in the face, it’s impossible not to feel connected to them, based on our common belief in Jesus Christ.
After all that, I understand completely Pope Francis is getting at when he calls us to an ecuмenism of blood. They died for the same Lord that I love, with all my heart, mind, and soul (at least to the best of my ability in my human frailty). They are the martyrs of all of us, and they matter to us, no matter what Christian faith we follow.
Ecuмenism of blood is the perfect term, and I’m meditating a lot on that as our country celebrates its Independence Day. I am, for the most part, free to practice my own faith in my own home and church – at least for the time being.
The Christian martyrs were not.
I can only hope that I could be as courageous as they if I’m ever asked to undergo the same suffering.

http://www.patheos.com/blogs/margefenelon/2015/07/the-ecuмenism-of-blood-pope-francis-scandalous-words-about-christian-martyrs/

This is the modern ecclesiology. The Catholic Church is reduced to the "fullness of the truth" but not necessary. You can be saved while only having a partial truth, it's just more difficult. So the necessity of belonging to the Church becomes one of precept rather than means. Heretics and schismatics belong to the Church by implicit desire if they are nice people you can get along with. A heretic or schismatic who dies for his faith is just as much a martyr as one who dies for the Catholic faith.

For now I will ask.  Is it possible for one who rejects Christ to have a supernatural Faith?

St. Augustine:

The Holy Ghost is to the Body of Christ, which is the Church, what the human soul is to the human body. It is by the soul that each member of the body lives and acts. In like manner, it is by the Holy Ghost that the just man lives and acts. As the soul does not follow a member which is cut off from the body, so, in like manner, does the Holy Ghost not follow a member which has been justly cut off from the Body of Christ. He, therefore, who wishes to obtain life everlasting, must remain vivified by the Holy Ghost; and in order to remain vivified by the Holy Ghost we must keep charity, love the truth, and desire unity.

This shows what I have said elsewhere, which is that the soul of the Church does not wander about outside of the Church like a disembodied ghost. The soul of the Church informs the body of the Church like the soul of the man informs the body of the man. If you want to belong to the soul of the Church, you have to belong to the body, because the soul is not to be found outside the body.

Of course, we would expect this from St. Augustine, because he is the most thorough defender of what I have called the ancient ecclesiology.

St. Augustine:

Quote from: St. Augustine

The Holy Ghost is to the Body of Christ, which is the Church, what the human soul is to the human body. It is by the soul that each member of the body lives and acts. In like manner, it is by the Holy Ghost that the just man lives and acts. As the soul does not follow a member which is cut off from the body, so, in like manner, does the Holy Ghost not follow a member which has been justly cut off from the Body of Christ. He, therefore, who wishes to obtain life everlasting, must remain vivified by the Holy Ghost; and in order to remain vivified by the Holy Ghost we must keep charity, love the truth, and desire unity.

This shows what I have said elsewhere, which is that the soul of the Church does not wander about outside of the Church like a disembodied ghost. The soul of the Church informs the body of the Church like the soul of the man informs the body of the man. If you want to belong to the soul of the Church, you have to belong to the body, because the soul is not to be found outside the body.

Of course, we would expect this from St. Augustine, because he is the most thorough defender of what I have called the ancient ecclesiology.

Pope Pius XII, in defining the Church as the Mystical Body of Christ, said:

"14. That the Church is a body is frequently asserted in the Sacred Scriptures. "Christ," says the Apostle, "is the Head of the Body of the Church."[13] If the Church is a body, it must be an unbroken unity, according to those words of Paul: "Though many we are one body in Christ."[14] But it is not enough that the Body of the Church should be an unbroken unity; it must also be something definite and perceptible to the senses as Our predecessor of happy memory, Leo XIII, in his Encyclical Satis Cognitum asserts: "the Church is visible because she is a body.[15] Hence they err in a matter of divine truth, who imagine the Church to be invisible, intangible, a something merely "pneumatological" as they say, by which many Christian communities, though they differ from each other in their profession of faith, are untied by an invisible bond.

And from the same encyclical:

Actually only those are to be included as members of the Church who have been baptized and profess the true faith, and who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed. "For in one spirit" says the Apostle, "were we all baptized into one Body, whether Jєωs or Gentiles, whether bond or free."[17] As therefore in the true Christian community there is only one Body, one Spirit, one Lord, and one Baptism, so there can be only one faith.[18] And therefore, if a man refuse to hear the Church, let him be considered - so the Lord commands - as a heathen and a publican. [19] It follows that those who are divided in faith or government cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit.

Take the Islamic ѕυιcιdє-bomber.

He has a sincere belief in "God as a Rewarder".
His conscience, in sincerity, tells him to kill the infidels according to what he believes to be God's commands.
He may be wrong about God's command, but he is invincibly ignorant. He was brainwashed by radical jihadis.
He is sorry for every sin he has committed against God, but thinks that killing infidels is holy, not a sin.

Now he goes and blows himself up in a Catholic school, killing 30 children and a teacher.

Is he saved by implicit faith and invincible ignorance? Is he a martyr for his faith?
If not, why not?

From what we as Catholics know (because it has been revealed by God), this man is not saved for he has not been incorporated into the Holy Roman Catholic Church by sacramental Baptism nor does he have an Explicit Faith in the Incarnation and the Holy Trinity, which are necessary, as a necessity of means, for salvation. This man is not a member therefore of the Mystical Body of Christ, which alone goes to Heaven.