What is there to motivate the search for these "extrinsic indicators of illegitimacy," if not the fact that the putative Pope has been teaching erroneous doctrine? And what would be the benefit of finding such indicators, if their presence leads one to conclude the very same thing he would have concluded in their absence, and from the very same motivations? And what is the point of a divine guarantee of indefectibility which can to all appearances be subverted by something as mundane as the blackmailing of a Pope, to the great detriment of souls, but which supposedly leaves the treasured papal prerogatives, which even the gates of hell cannot prevail against, nevertheless undiminished?
It seems you are just making extra work for yourself, Ladislaus. And you still haven't escaped the spectre of private judgment, at least not as you define it.
Can we perhaps just cut the Gordian Knot here and admit that "private judgment" does not mean what you think it means? When someone compares the teachings of the pre-Conciliar Popes to those of the post-Conciliar Popes and finds that they are not the same, that is not a willful and sinful use of private judgment in a religious matter, but plain and simple logic of the sort used every day. Furthermore, the modus tollens argument from within the scope of the magisterium is the normal and ordinary means used to expose heresies and heretics of every kind; and if those heretics happen to be putative Popes, then so much the worse for them.
As I've pointed out to you before, your brand of dogmatic sededoubtism is very, very dogmatic indeed (in the literal sense of the word), for it casts a pale of uncertainty over every papal proposition whatsoever, which uncertainty can never be relieved until the end of time. Sededoubtism equates to eschatological skepticism with a proviso attached to the effect that only he who perseveres in unyielding recognition of the papal claimant, even in cases of heresy, will be saved. The Catholic Church has never taught such doctrine.
Now here is the whole argument expressed in the simplest possible terms. Both Bishop Sanborn and Bishop Williamson agree that what Francis teaches isn't Catholic. In this they are not exercising their private judgment but are simply being rational. +Williamson recognizes Francis as Pope but then proceeds to sift his teachings for what is traditional in them. +Sanborn does not recognize Francis and does not sift his teachings. The accusation of magisterium-sifting by private judgment pertains to +Williamson precisely because he recognizes Francis as Pope, as the infallible head of the magisterium whom he is required to obey. The accusation of pope-sifting by private judgment does not pertain to +Sanborn because there is no Catholic teaching which asserts that someone who claims to be a Pope necessarily is one. In fact, the Catholic Church teaches that heretics are cut off from the Body of Christ. He who is not a member of the body cannot be the head of the body. Therefore, Francis is not the Pope.
Whether or not you accept sedevacantism, you must accept that +Williamson's magisterium-sifting involves private judgment in a way that +Sanborn's alleged "pope-sifting" does not. That is the only point being hashed out here. Everything which follows from this is an implication which I will not belabor at present. In no wise, however, can the two cases be treated as equivalent, nor does the sededoubtist position hold water. It all boils down to the major premise of +Bishop Sanborn's syllogism which, if memory serves me, has not even been mentioned in this thread, viz. that Vatican II introduced substantial changes in Christian doctrine. That is the stone which grinds everything else to dust. If that is true than you should be able to draw the conclusions as well as I.