Nicely put. Maybe it helps to think of the purpose of those conversations where pointing out logical fallacies might help? Such conversations are almost always

Nicely put. Maybe it helps to think of the purpose of those conversations where pointing out logical fallacies might help? Such conversations are almost always about changing someone's view of the world - sometimes through debate, where conflicting positions are taken and argued upon, and sometimes simply because someone (e.g. you) asks for a clarification in order to know which position to take. So I think that the idea is that people can at least take well-reasoned positions, not necessarily equal positions, in a world that is not black and white. The list of fallacies can then be a useful guide to judge the persuasiveness of the arguments of the different sides, including of course your own arguments, and to help you see whether a position might need to change because it is not well-reasoned. In your example, I can say "I trust Fields Medallists" and therefore judge the proof to be likely wrong. That is a valid position to take. You may say that "the author of the proof is a very talented mathematician" so the proof is likely right, which would be equally valid. What we cannot do, without further information, is to claim that the proof must with absolute certainty be right or wrong - that I believe would be the fallacy here.