It might not be of interest to cutting-edge researchers, but I think it's still important from a pedagogical perspective to see what mistakes are made by intelligent, educated, non-crackpots who think they've found a glaring error, or some fantastical consequence.

Usually it would be some common mistake/misconception, eg. applying special relativity in a non-intertial reference frame, but occasionally it might give a real insight into the analogies and methods used to teach a concept.

Yeah, probably. I just like it more when wrong people are angry. Calm, reasonable but also wrong people make me uneasy.

Say that you encounter a conspiracy theorist. When you point out that his theory is crazy, you don't want him to propose to agree to disagree, right? At that point, I feel like I might be wrong, because I'm the most argumentative. Get what I mean?