Yet doesn't Fr. Chazal insist that they are fully valid and true popes?
Because he has no authority to say otherwise? I think so, yes.
Where does he get the authority to reject the New Mass and VII?
At the point they break with Tradition, and, therefore place our faith in danger. Needing to determine the validity of one's office does not have an impact, one way or another, on our faith.
It's quite simple. The moment someone tells me to do something which is contrary to the Faith, I must reject it. To feel compelled to take it a step further and find out if they are what they say they are, to go out of my way to find out what the canons, theologians, et al., think is possible, when it has no real impact on my following of the Faith (as I hope I've just shown in the prior sentences is possible (and necessary) to do). His holding of his office has no bearing on my holding of my Faith then.
Bishop WIlliamson puts it much clearer, I think. (Emphasis mine):
The question is not of prime importance. If they have not been Popes, still the Catholic Faith and morals by which I must “work out my salvation in fear and trembling” (Phil. II, 12) have not changed one iota. And if they have been Popes, still I cannot obey them whenever they have departed from that Faith and those morals, because “we ought to obey God rather than men” (Acts, V, 29). However I believe in offering answers to some of the sedevacantists’ arguments, because there are sedevacantists who seem to wish to make the vacant See of Rome into a dogma which Catholics must believe. In my opinion it is no such thing. “In things doubtful, liberty” (Augustine).
What are your thoughts, my friend?
My thoughts are concurrent with your first sentence. Man's first duty is to follow the true religion, and it follows that he also has a duty to refuse a false religion.
The true religion is exercised and taught by the Catholic Church (to the exclusion of all other Churches), so it is also the onus of Christians to belong to it, and it therefore necessarily follows that they submit themselves to the authority of that same Church.
Anyone who has been following my posts knows that I have historically had a pretty
laissez faire attitude toward one's approach to the crisis, so long as their traditionalism is manifested by their rejection of the N.O. program. However, I am no longer convinced, at least at this very moment, that remaining in the R&R camp has
no impact on one's faith.
If the pope of the Catholic Church "declares and defines... for the whole Church... by the authority of Jesus Christ, the apostles Peter and Paul and our own..." A, B, C or D, then he is speaking with the full power of his office, and defining something for the whole Church.
(This is distinct from the Church's infallibility in canonizations)
If this is not an infallible declaration, nothing is. Arguments about process or some other mitigating factor reduces infallibility to the human phenomenon of "being right." It is no longer the Holy Ghost which guides the Pope (or the Church); the Holy Ghost guides individuals in determining what teachings of the Church will lead them to Heaven and which will lead them to Hell.
Beginning every query related to the traditional faith with the dual premise that sedevacantism is false and that the Church can teach any manner of error so long as it doesn't go contrary to what the SSPX believes about ecclesiology, authority and infallibility IS harmful to the faith. There is no principle or theological guide by which a R&R Catholic MUST accept ANY teaching of the Catholic faith if Francis can declare and define, for the whole Church, by the authority of Jesus Christ, the Holy Apostles Peter and Paul and HIS OWN A, B or C and not be protected by the Holy Ghost in doing so.
An R&R Catholic cannot even argue that Catholics must attend the TLM (vs. the N.O.) or abide by the teachings of Pius X (or pick any other pope) without contradicting his own position.