The major problem is the denial of EENS, that is what is heretical..not so much BOD theories (although the simple fact of the matter is that the fruit of BOD tolerance is near universal EENS denial). What seemingly 99% of trad clerics believe is that Jews, Hindus, Muslims, etc. can be saved while being Jews, Hindus, Muslims, etc. That they can be saved in these false religions. Yes, they say "but not by those religions". They say those non-Catholics can be united to the "soul of the Church", but not the Body. It is all contrary to what has already been taught.
This is just one example:
Pope Eugene teaches the Dogma exactly as it is to be believed.
"Pagans, Jews, heretics, and schismatics" are outside of the Church.
Pope Eugene IV makes no exceptions. Pagans, Jew, heretics, and schismatics are outside of the Church. Nice ones, mean ones, just-plain-evil ones, keep-the-natural-law good ones, are all outside of the Church, not joined to the Church, not abiding in it, not in the Church's bosom, not in the unity of the ecclesiastical body.
As Pope Eugene IV taught this from the Chair of Peter, we are bound to believe this as Divinely revealed
Again (and I think this is a point that cannot be stressed enough) Pope Eugene IV, speaking from the Chair of Peter, teaches us that pagans, Jews, heretics and schismatics are outside the Church, not joined to the Church, not abiding in the Church, not in the bosom of the Church, not in the unity of the ecclesiastical body, without exception
He does not say that a pagan, jew, etc. is outside, not joined, not abiding in, not in the bosom of, not in the unity of the ecclesiastical body now, but might not necessarily be later if they are "invincibly ignorant" or "follow the natural law". On the contrary, he teaches that every non-Catholic is outside, not joined, not abiding in, etc. the Church as long as are non-Catholic and non-members. An "invincibly ignorant" Hindu is a Hindu 1 decade before he dies, 1 year before, 1 day before, 1 hour before, 1 second before. If that Hindu is still in the Hindu religion when he dies (as these Trad clerics claim), then he falls under the "pagans" Pope Eugene IV referenced, and is outside of the the Church, and cannot be saved!
It is a clear denial of the Dogma for these Trad clerics to say that pagans, jew, heretics, and schismatics can be saved in their false religion by being "united" or "joined" to the "soul" of the Church, even if they try to save themselves by simultaneously saying that the non-Catholics are not saved by their false religion..because as has been taught, non-Catholics are outside, not joined to, not abiding in, not in the bosom of Church, not in the unity of the ecclesiastical body. Pope Eugene IV made clear that they are in no way "joined" or "united" to any "part" of the Church, at any time
This part is especially important, and is a clear refutation not only of the EENS-denial, but I think also of BOD/BOB:
The Council of Trent infallibly teaches that the sacraments are necessary for salvation, though not all. Meaning that at least one is. That one is, obviously, Baptism..which is the "gateway to the spiritual life", and "holds first place among all the sacraments"
The problem that not only EENS-deniers face, but also BOD proponents, is that those who have not received the Sacrament of Baptism are not in the unity of the ecclesiastical body (i.e. members of the body of Christ, the Church). No theologian has ever taught that BOD makes someone an actual member of the Church, as that would be contrary to infallible Church teaching (Trent Sess 14, Ch. 2, Eugene IV at Florence, Exultate Deo), teaching that was also echoed by Pius XII (Mystici Corporis, Mediator Dei), I'm sure other popes as well, but I would have to look around. Only the Sacrament of Baptism, celebrated in water, makes someone a member of the Body of Christ.
So for the EENS deniers and BOD proponents, the problem is that Eugene IV infallibly teaches that pagans, jews, etc are in no way "united" or "joined" to the Church, and that only for those in the unity of the ecclesiastical Body do the Sacraments contribute to salvation. Those same sacraments that are necessary for salvation. Those pagans, Jews, etc. are not members of the Body, so the Sacraments cannot contribute to their salvation. The same Sacraments that are necessary for salvation!
What you say is absolutely true for "salvation," which means going straight to Heaven without a stop in any other abode after death. No one, who is not a baptized member of the Church has the chance "to benefit from the ecclesiastical sacraments and fasting, almsgiving and other offices of piety and exercises of the Christian soldiery that bring forth eternal rewards." These things prepare the white garment of the soul. And they are only available to Catholics.
However, not all who are outside the Church will burn in Hell forever and never make it to Paradise. In fact, even most Catholics will suffer "the fires of Hell" in Purgatory for a long time before entering Paradise.
The point is that there are other temporary places for those non-saved souls are "detained," and we shouldn't lose sight of that fact. Aquinas speaks of five places:
https://aquinas.cc/la/en/~ST.IIISup.Q69.A7I answer that, The abodes of souls are distinguished according to the souls’ various states. Now the soul united to a mortal body is in the state of meriting, while the soul separated from the body is in the state of receiving good or evil for its merits; so that after death it is either in the state of receiving its final reward, or in the state of being hindered from receiving it. If it is in the state of receiving its final retribution, this happens in two ways: either as to good, and then it is
paradise; or as to evil, and thus as regards actual sin it is
hell, and as regards original sin it is
the limbo of children. On the other hand, if it be in the state where it is hindered from receiving its final reward, this is either on account of a defect of the person, and thus we have
purgatory, where souls are detained from receiving their reward at once on account of the sins they have committed, or else it is on account of a defect of nature, and thus we have
the limbo of the fathers, where the fathers were detained from obtaining glory on account of the guilt of human nature, which could not yet be expiated.
Here is what Aquinas says about the Limbo of Hell (aka the limbo of the fathers, or Abraham's Bosom), which is where those who were "just according to the natural law" ended up:
https://aquinas.cc/la/en/~ST.IIISup.Q69.A4I answer that, After death men’s souls cannot find rest save by the merit of faith, because he that comes to God must believe(Heb 11:6). Now the first example of faith was given to men in the person of Abraham, who was the first to sever himself from the body of unbelievers, and to receive a special sign of faith: for which reason the place of rest given to men after death is called Abraham’s bosom, as Augustine declares (On the Literal Meaning of Genesis12).
But the souls of the saints have not at all times had the same rest after death; because since Christ’s coming they have had complete rest through enjoying the vision of God, whereas before Christ’s coming they had rest through being exempt from punishment, but their desire was not set at rest by their attaining their end. Consequently, the state of the saints before Christ’s coming may be considered both as regards the rest it afforded, and thus it is called Abraham’s bosom, and as regards its lack of rest, and thus it is called the limbo of hell.
Accordingly, before Christ’s coming the limbo of hell and Abraham’s bosom were one place accidentally and not essentially: and consequently,
nothing prevents Abraham’s bosom from remaining after Christ’s coming and from being altogether distinct from limbo, since things that are one accidentally may be parted from one another.