"Consequently, the Society of St. Pius X, its priests, brothers, sisters, and oblates, cannot tolerate among its members those who refuse to pray for the Pope or affirm that the Novus Ordo Missae is per se invalid.

The foundational theological error of the sspx, both under +ABL and now, is the following:
Validity of the new mass = ok to attend (i.e. moral and pleasing to God).
.
Even if the new mass is valid, it is still illict and immoral, thus it is gravely sinful.  You could also argue that +ABL is wrong to assume validity, when Cardinals Ottaviani, Bacci and others said that one could POSITIVELY DOUBT such validity.  I just can't agree with +ABL on this wishy-washy view.
 

True, EENS-liberalism would be problem 1.  So the limp-wrist treatment of the new mass is #2.
.
I would give them some human understanding (but not forgiveness) for being liberal on #1, simply because this liberalism goes back to the 1800s and thus is quite pervasive and complex. 
.
Error #2 is blatant, in-your-face novelty, and +Ottaviani minced no words in condemning it, neither did MANY early Trad pillars like Fr Depauw, Fr Wathen, and many other French monastics (who I forget their names, but who were close with +ABL).  So for the sspx to take such a lukewarm attitude even in the 70s/80s (when the craziest, most outlandish and most blasphemous new masses were going on), is quite sad. 

I provided the quote from Archbishop Lefebvre's 1979 statement to demonstrate the attitude the Archbishop publicly professed towards Sedevacantism at this time, as I thought it was relevant to the thread. Not surprisingly it has reignited debate about the Pope, the NOM and the position of the Archbishop.

I think the Archbishop's statement needs to be viewed in light of Fr Chazal's affirmation in Contra Cekadam (p55):
"All the while, the Archbishop was so convinced that the situation was so difficult to solve that he collaborated, even after 1983, with a long list of non dogmatic sedevacantist priests: Fathers Vinson, Sochel, Schaeffer, Mouraux, Coache, Raffali and others I can't recall. Like him, they too probably understood that the situation is not so simple. These priests were in good standing, despite their views differing from his, and applied discretion. Hence, likewise, he wouldn't touch what still stood. it is a great and wise canonical principle."

I am quite certain that no Bugnini transitional Mass was ever used at Econe. I wonder if Ladislaus is not thinking of a certain letter he has read addressed by Fr de Lauriers to the Archbishop accusing him of saying the "New Mass", which Mass was not the NOM as one might mistakenly be led to believe, but the 1962 Missal currently in use in the SSPX.

I think it not justified to accuse the Archbishop of a 'wishy washy' attitude towards the New Mass. Here follows the full statement from 1979 as clear enough evidence to the contrary. If one knows anything about the Archbishop and has read his books and many statements and conferences on this subject, I don't see how one can maintain such an assertion in truth and justice.

8 November 1979

How often during these last ten years have I not had occasion to respond to questions concerning the weighty problems of the New Mass and the Pope. In answering them I have ever been careful to breathe with the spirit of the Church, conforming myself to her Faith as expressed in her theological principles, and to her pastoral prudence as expressed in moral theology and in the long experiences of her history.

I think I can say that my own views have not changed over the years and that they are, happily, those of the great majority of priests and faithful attached to the indefectible Tradition of the Church.

It should be clear that the few lines which follow are not an exhaustive study of these problems, The purpose, rather is to clarify our conclusions to such an extent that no one may be mistaken regarding the official position of the Society of St, Pius X.

It must be understood immediately that we do not hold to the absurd idea that if the New Mass is valid, we are then free to assist at it. The Church has always forbidden the faithful to assist at the Masses of heretics and schismatics, even when they are valid. It is clear that no one can assist at sacrilegious Masses or at Masses which endanger our faith.

Now, it is easy to show that the New Mass, as it was formulated by the officially authorized Conciliar Liturgical Commission considered together with the accompanying explanation of Mgr. Bugnini, manifests an inexplicable rapprochement with the theology and liturgy of the Protestants. The following fundamental dogmas of the Holy Sacrifice of the Mass are not clearly represented and are even contradicted:

It is sufficient to enumerate a few of the novelties in the New Mass to be convinced of the rapprochement with the Protestants;

All these innovations are authorized. One can fairly say without exaggeration that most of these Masses are sacrilegious acts which pervert the Faith by diminishing it. The de-sacralization is such that these Masses risk the loss of their supernatural character, their mysterium fidei; they would then be no more than acts of natural religion. These New Masses are not only incapable of fulfilling our Sunday obligation, but are such that we must apply to them the canonical rules which the Church customarily applies to communicatio in sacris with Orthodox Churches and Protestant sects.

Must one conclude further that all these Masses are invalid? As long as the essential conditions for validity are present (matter, form, intention, and a validly ordained priest), I do not see how one can affirm this.

The prayers at the Offertory, the Canon, and the Priest’s Communion which surround the words of Consecration are necessary, not to the validity of the Sacrifice and the Sacrament, but rather to their integrity. When the imprisoned Cardinal Mindszenty, desiring to nourish himself with the Body and Blood of Our Lord, and to escape the gaze of his captors, pronounced solely the words of Consecration over a little bread and wine, he most certainly accomplished the Sacrifice and the Sacrament.

It is clear, however, that fewer and fewer Masses are valid these days, as the faith of priests is destroyed and they possess no longer the intention to do what the Church does – an intention which the Church cannot change. The current formation of those who are called seminarians today does not prepare them to celebrate Mass validly. The propitiatory Sacrifice of the Mass is no longer considered the essential work of the priest. Nothing is sadder or more disappointing than to read the sermons or teachings of the Conciliar bishops on the subject of vocations, or on the occasion of a priestly ordination. They no longer know what a priest is.

Nevertheless, in order to judge the subjective fault of those who celebrate the New Mass as of those who attend it, we must apply the roles of the discernment of spirits given us in moral and pastoral theology. We (the priests of the Society) must always act as doctors of the soul and not as judge and hangmen. Those who are tempted by this latter course are animated by a bitter spirit and not true zeal for souls. I hope that our young priests will be inspired by the words of St. Pius X in his first encyclical, and by the numerous texts on this subject to be found in such works as The Soul of the Apostolate by Dom Chautard, Christian Perfection and Contemplation by Garrigou-Lagrange, and Christ the Ideal of the Monk by Dom Marmion.

Let us now pass to a second but no less important subject: does the Church have a true Pope or an impostor on the Throne of St. Peter? Happy are those who have lived and died without having to pose such a question! One must indeed recognize that the pontificate of Paul VI posed, and continues to pose, a serious problem of conscience for the faithful. Without reference to his culpability for the terrible demolition of the Church which took place under his pontificate, one cannot but realize that he hastened the causes of that decline in every domain. One can fairly ask oneself how it was possible that a successor of Peter can, in so little time, have caused more damage to the Church than the French Revolution.

Some precise facts, such as the signatures which he gave to Article VII in the Instruction concerning the New Mass, and to the Declaration on Religious Liberty, are indeed scandalous and have led certain traditionalists to affirm that Paul VI was heretical and thus no longer Pope. They argue further that, chosen by a heretical Pope, the great majority of the cardinals are not cardinals at all and thus lacked the authority to elect another Pope. Pope John Paul I and Pope John Paul II were thus, they say, illegitimately elected. They continue that it is inadmissible to pray for a pope who is not Pope or to have any "conversations" (like mine of November 1978) with one who has no right to the Chair of Peter.

As with the question of the invalidity of the Novus Ordo, those who affirm that there is no Pope over-simplify the problem. The reality is more complex. If one begins to study the question of whether or not a Pope can be heretical, one quickly discovers that the problem is not as simple as one might have thought. The very objective study of Xaverio de Silverira on this subject demonstrates that a good number of theologians teach that the Pope can be heretical as a private doctor or theologian but not as a teacher of the Universal Church. One must then examine in what measure Pope Paul VI willed to engage in infallibility in the diverse cases where he signed texts close to heresy if not formally heretical.

But we can say that in the two cases cited above, as in many another, Paul VI acted much more the Liberal than as a man attached to heresy. For when one informed him of the danger that he ran in approving certain conciliar texts, he would proceed to render the text contradictory by adding a formula contrary in meaning to affirmations already in the text, or by drafting an equivocal formula. Now, equivocation is the very mark of the Liberal, who is inconsistent by nature.

The Liberalism of Paul VI, recognized by his friend, Cardinal Daniélou, is thus sufficient to explain the disasters of his pontificate. Pope Pius IX, in particular, spoke often of the Liberal Catholic, whom he considered a destroyer of the Church. The Liberal Catholic is a two-sided being, living in a world of continual self-contradiction. While he would like to remain Catholic, he is possessed by a thirst to appease the world. He affirms his faith weakly, fearing to appear too dogmatic, and as a result, his actions are similar to those of the enemies of the Catholic Faith.

Can a Pope be Liberal and remain Pope? The Church has always severely reprimanded Liberal Catholics, but she has not always excommunicated them. Here, too, we must continue in the spirit of the Church. We must refuse Liberalism from whatever source it comes because the Church has aways condemned it. She has done so because it is contrary, in the social realm especially, to the Kingship of Our Lord.

Does not the exclusion of the cardinals of over eighty years of ages, and the secret meetings which preceded and prepared the last two Conclaves, render them invalid? Invalid: no, that is saying too much. Doubtful at the time: perhaps. But in any case, the subsequent unanimous acceptance of the election by the Cardinals and the Roman clergy suffices to validate it. That is the teaching of the theologians.

The visibility of the Church is too necessary to its existence for it to be possible that God would allow that visibility to disappear for decades. The reasoning of those who deny that we have a Pope puts the Church in an inextricable situation. Who will tell us who the future Pope is to be? How, as there are no Cardinals, is he to be chosen? This spirit is a schismatical one for at least the majority of those who attach themselves to certainly schismatical sects like Palmar de Troya, the Eglise Latine de Toulouse, and others.

Our Fraternity absolutely refuses to enter into such reasonings.

We wish to remain attached to Rome and to the Successor of Peter, while refusing his Liberalism through fidelity to his predecessors. We are not afraid to speak to him, respectfully but firmly, as did St. Paul with St. Peter.

And so, far from refusing to pray for the Pope, we redouble our prayers and supplications that the Holy Ghost will grant him light and strength in his affirmations and defense of the Faith.

Thus, I have never refused to go to Rome at his request or that of his representatives. The Truth must be affirmed at Rome above all other places. It is of God, and He will assure its ultimate triumph.

Consequently, the Society of St. Pius X, its priests, brothers, sisters, and oblates, cannot tolerate among its members those who refuse to pray for the Pope or affirm that the Novus Ordo Missae is per se invalid. Certainly, we suffer from this continual incoherence which consists in praising all the Liberal orientations of Vatican II and at the same time straining to mitigate its effects. But all of this must incite us to prayer and to the firm maintenance of Tradition rather than to the affirmation that the Pope is not the Pope.

In conclusion, we must have that missionary spirit which is the true spirit of the Church. We must do everything to bring about the reign of Our Lord Jesus Christ according to the words of our Holy Patron, St. Pius X: Instaurare omnia in Christo. We must restore all things in Christ, and we must submit to all, as did Our Lord in His Passion for the salvation of souls and the triumph of Truth. "In hoc natus sum," said Our Lord to Pilate, "ut testimonium perhibeam veritati."

“I was born to give witness to the Truth."

A year later, he acknowledged that, although the NOM could not be the object of an obligation to attend, nevertheless, those who were scrupulous could still fulfill their Sunday obligation by attending it (Apologia Pro, Vol. II, Ch 40):

"Unfortunately, the Archbishop's statement was not as clearly worded as it might have been on the matter. One passage in particular gave some readers the impression that the Archbishop had stated that a Catholic could never fulfill his Sunday obligation by assisting at the New Mass. Among those who had received this impression from the statement was Cardinal Seper, who mentioned the anxiety it had caused him during an interview he granted me at Easter 1980. I had the opportunity of a long interview with the Archbishop a few weeks later when we discussed the matter. He was kind enough to summarize his considered opinion for me in writing (dated 9 May 1980). It read as follows:

Quote Those who feel themselves obliged in conscience to assist at the New Mass on Sunday can fulfil their Sunday obligation. But one cannot accuse a person of a grave fault because he prefers not to assist at Mass on Sunday rather than assist at the New Mass.

Thus where the Archbishop states that “these New Masses are incapable of fulfilling our Sunday obligation,” he is referring to New Masses which involve “sacrilegious acts which pervert the faith by diminishing it.” The declaration which he made at my request makes it quite clear that this was indeed his meaning."
https://www.sspxasia.com/Docuмents/Archbishop-Lefebvre/Apologia/Vol_two/Chapter_40.htm

Pax, how do you reconcile that the Church could give an “illicit and immoral” mass with this condemnation from the Council of Trent: “ Canon 7. If anyone says that the ceremonies, vestments, and outward signs which the Catholic Church uses in the celebration of masses, are incentives to impiety rather than stimulants to piety,[26] let him be anathema.“?

1.  The new mass is illicit, immoral and very, very probably invalid.  You didn't mention invalidity, so I wanted to make sure this isn't forgotten.  
2.  The "new mass" isn't a mass at all, so your canon doesn't apply.  The new mass is a eucharistic memorial, a communion meal, a Holy Thursday remembrance.  No where in the new mass is there an offering, an offeror, a victim or a sacrifice offered to God.  So it's not a sacrifice, hence it's not a mass.
3.  The Church did not "give" us this new mass, because it did not come from the Apostolic Powers of the pope.  If you read Paul VI's constitution which created the novus ordo, he says that he is creating this rite by order of the Vatican 2 council.  He never created it by order of his Apostolic authority, so it is not a Traditional rite.
4.  Even if Paul VI did create this rite using his papal powers, he did not order anyone to use it, nor is there a penalty for not using it.  All he did was create it.  So, the Church did not "give" us a new rite, in a strict legal sense.  The Church created a new rite, but the creation of something is different than the use of it.  Creating the atomic bomb is not necessarily wrong, for this is just an advance of technology and a discovery of atomic power.  But USING the atomic bomb is wrong, except for extreme circuмstances (and that's even debatable).  
5.  Quo Primum is the law on the books, both legally and morally, for the Latin rite.  There's nothing preventing a bad pope from passing illegal laws or immoral laws.  It's the obligation of Catholics to know their faith and to pray for guidance when they are confused.  God will guide those of good will, as He did when people created the Traditional movement and rejected the evils of the new liturgy.  Quo Primum was printed in every missal for centuries, and for good reason.  When the new mass came out, people knew it was not the "old mass".  They knew it was wrong.  They went along anyway.  They were not forced (except by their bishops and priests), but such force didn't come from rome.  They accepted the new liturgy because it was easier.