Certainly doesn't answer the question, but seems at least tangentially relevant: all modular representations are a direct summand of a module that admits a finite resolution by permutation modules arxiv.org/abs/2003.04373.

@Kostya_I: I agree these are all closely related. If one question's answers could all be included as answers to another question - but the converse is not true - is it standard for the "sub-question" to be marked as duplicate?

@Kostya_I: I don't think so, although there is obviously some overlap and some answers would work for both. For example, some highly upvoted answers there include the Grothendieck prime and the fact that Cayley didn't realize $C_2 \times C_3 \cong C_6$, but those aren't relevant to my question.

@NeilHoffman: Not quite. That was actually Francois Ziegler's original suggestion, which my comment 2 above was in response to.

@FrancoisZiegler: Maybe, but those are more along the lines of vague-but-essentially-correct intuitions that then became more formalized (which I think is how most math goes), whereas I'm interested in more extreme examples of vague-but-more-incorrect ideas that still led to good math in the end.

@LSpice: What if I just remove "BS" from the title & question body?

Also, if it is a famous open problem, it'd be nice to have at least one reference to where it's been discussed. I've found it not so easy to search for... @MarkSapir do you know one?

I think it's worth pointing out that the linked Etingof-Gelaki paper doesn't just give such examples, it gives a complete characterization of when two non-isomorphic groups can have equivalent tensor categories (and, there, non-equivalent symmetric tensor categories).