Sedevacantists don't say that one must accept and do everything the pope does, either. It's just that in this case, it isn't a matter of simply not accepting everything... traditional Catholics don't accept anything(!) from the conciliar claimants. In fact, they positively reject everything they do.
You've lost my point.
There are very specific conditions for speaking ex cathedra. The Liberals absolutely ignore this. For example, in an off the cuff remark Bergoglio says that religious liberty is true Christian teaching. He is spouting personal erroneous opinions and material heresies, but he is not speaking in an infallibly and binding way. Fr. Hesse explains it best. It is pure papolotry to hold Catholics as bound to every opinion of someone one believes to be (maybe) the pope.
If that is your point, I fail to see its relevance: sedevacantists don't think that the post-conciliar popes aren't popes because they taught heresy ex cathedra. After all, the very notion of such an instance is absurd, and no theologian admits the possibility.
Sedevacantists think as they do because the post-conciliar claimaints are manifest heretics.
The significance of Bergoglio "spouting" "personal heresy" is greater than you think. He needn't "teach" it "as pope". He can't, that's an impossibility. Him manifesting his heresy (which is what he does when he "spouts", as you say) is sufficient enough to show that he is not a Catholic, and cannot be pope.
This is beyond what Fr. Berry is saying in the quote of his, but this is the fundamental principle of the sedevacantist position. The post-conciliar claimants are manifest heretics, and therefore, not members of the Church, and certainly not its head.
I would also like to know where in Catholic teaching does it state that it is a mortal sin for anyone to doubt the legitimacy of a papal claimant. There are saints that held to rival claimants during the Western Schism. This is another novelty.
It doesn't. His claim that there is some "objective grave sin" in the matter is spurious. The entire concept of "objective sin" as commonly applied by traditionalists today (e.g., "objectively sinful to attend the new mass") is somewhat smoke and mirrors. Sin requires an act of the will. One cannot inadvertently offend God.
And then you state that your reject anything and everything of the papal claimants. This, of course, is more exaggeration. Everything? What if he says that Islam is not a religion of peace? Do you reject this? What do you mean that you reject anything and everything the papal claimant says? Even if it is obviously correct?
Centroamerica, if you happen to "accept" when Francis says the sun is shining, that's not evidence of a relationship between a Catholic and the pope. Or when he states some obvious tenet of the Christian tradition, like Jesus having had a mother named Mary.
The relationship between a Catholic and the pope is one of obedience and one of learning. Most of us haven't lived with an actual pope so it's difficult to understand without praxis, but if you look at Church history and look at how Catholics treated the pope, it becomes easier.
Name one thing that Francis has taught that you accept on his authority as pope. While you go off searching for whatever that is, ask yourself why you have to Google something the pope has said that you agree with, and why you can't simply tell me what you've learned from him. This very behavior is evidence that the relationship you have with him is one of suspicion and scrutiny, not deference, obedience, and learning.
Don't confuse what I just said as an argument for him not being pope. It's simply an exercise to show that you certainly don't
treat him as pope in any meaningful way. And you should ask yourself why. If the answer is "because he's a heretic" or "because he doesn't teach anything anyways" then the dilemma just becomes deeper, not solved. It raises more questions than it answers.