One of the tricks of TOFP is to make "sedevacantism" into a single theory, and then rebut all of the worst arguments, creating the impression that this proves that Paul VI or Francis, etc., have been true popes. TOFP just picks the low-hanging fruit and pretends that the tree is then bare.
Of course, consistency isn't their strong point, so they also make whatever hay they can out of the reality that sedevacantists have different views on some pretty fundamental things, so they employ one sede against another, and show in this way that their invented, non-existent "sect" is a house divided. As if sedeplenists all have the same theory, especially sedeplenists like Hans Kung and Robert Siscoe, just to pick two very strikingly different sedeplenist theorists...
The truth is, there's no sede sect (although there may be sects which are sede... yes, I'm thinking of you, Dimond Brethren).
All right, now to the substance of the matter.
TOFP ignores a lot of what the Church teaches about her own magisterium, which is the first thing any serious person would want to know, and to get onto the page, before assessing the facts in order to work out what's happened.
So, what is the relevant doctrine?
First, that the Church when she teaches does not offer advice, she imposes her doctrine authoritatively. Importantly, there is no special form of words required to indicate this imposition of doctrine - when the Church speaks, she speaks as one having authority. She does not need to repeat, every time, that she means what she says.
Second, as a result of the authoritative nature of the magisterium, the faithful are obliged to accept what comes from the Church. This obligation involves changing one's mind, if necessary, so as to give sincere internal assent to the doctrine proposed. This is irrespective of the fact that the doctrine is true. We're not obliged to give assent because it's true, but because the Church is commissioned by God to teach us.
Third, the Church is aided by the Holy Ghost so that she is the sure ark of salvation, only teaching what is contained in, or compatible with, the deposit of faith. This is not to say that every teaching act is infallible, but that what comes from the magisterium is infallibly safe. So, whatever error might conceivably come from the Holy See, it won't be dangerous to the faith.
Now, infallibility. This is a much-abused notion. In brief, the Church is infallible, so we must identify when it is the Church herself teaching, as opposed to some representative of the Church teaching without committing the Church herself to his words. We want to know, has the Church committed herself to this doctrine? If so, she cannot have erred. If not, then some error is possible. The Church commits herself either through the Roman Pontiff acting alone, or through all of her ordinaries, under the Roman Pontiff, acting together. Again, whatever error might be present in a magisterial act, it won't be dangerous. An error of fact, an error of detail, but no heresy or theological error. When considering the pope heretic thesis, the theologians always restrict the possibility of a pope professing heresy to his private acts. This is why.
Having gotten all of this down (in summary), we can turn to the events of fifty years ago and examine them in its light.
Firstly, John XXIII introduced a new approach to what should have been authoritative proposing of doctrine. He eschewed the notion of canons (i.e. laws, doctrine in legal form) with anathemas, and suggested that the Church no longer teaches with authority, but merely offers advice.
Secondly, the language of Vatican II was journalistic, not authoritative, in line with this new philosophy.
Finally, the practical approach adopted was tyrannical, not according to the rule of law. So those who refused the new doctrines and the new liturgy were persecuted in various ways, generally without resort to proper legal processes. Virtually all appeals from the faithful, and the clergy, were ignored or sidelined.
The "snap!" answer of shallow thinkers is to point out that if the Church didn't commit herself to the new doctrines, then infallibility isn't at issue, end of story. This is essentially what Siscoe and Salza have done in TOFP. It hardly begins to deal with the problem.
The problem itself is best seen by pointing out that in fact, any individual who approaches the local "Catholic Church" in the West, and in a large part of the East, will be invited to believe things that are not only not true, but worse, are incompatible with the teaching of the Church; he will be invited to worship in a way that does not give due honour to God, and which trains him out of the Catholic Faith; in sum, he will not be put surely in the way of salvation, but instead he will be diverted from it. The results prove this, even it weren't obvious from examining the doctrines and the liturgy and the pastoral practices themselves.
Francis has now put this problem into neon lights for all, but it's a fifty-year-old problem.
A purely technical answer to the problem is to say that the clever Modernist revolutionaries side-stepped clearly infallible modes of teaching so as to avoid a clear collision with the dogma of infallibility, and so what has happened is that a great facade of pseudo-law, and pseudo-teaching, has been erected unlawfully, blinding the clergy and the faithful, without harming the essential attributes of the Church. If this were so, then the doctrine behind it would be something like, "The Church can lead people to hell in practice, but only by deceiving people into thinking that they must follow when actually, if they were clever and had their eyes open, they would have seen that they were being led into hell."
Do you believe in that Church? No, neither do I. So the problem remains.
In any case, it is hardly clear that the Roman Pontiff can validly cease teaching authoritatively, no matter what John XXIII tried to do. In other words, just as a pope cannot validly declare that priests can be made without ordination, neither can he decide that a general council can issue non-binding advice instead of binding doctrine.
The best answer, as you'd expect me to say, is that the See of Rome has been vacant, some of the bishops have continued to preach the true faith and maintain the Church's traditions (much more in the East than in the West, where only two bishops held fast), and many have not. There's nothing new in such a situation, except the duration and extent of the crisis. Without a Roman Pontiff the Church cannot commit herself to any new definition of doctrine, so the question is reduced to individual bishops acting on their own authority. You can still say, well how can a good God permit this?, and the answer is, His ways are not our ways. What He cannot do is go contrary to His own promises. He has promised that the Church will not err, that she will not give stones when her children ask for bread; He has not promised to prevent false teachers from arising who will deceive, if possible, even the Elect. In fact, He has warned us explicitly that this will happen...