> and it was only because people credibly believed he had one that they spent so much effort trying to (supposedly, re-)discover it.

That's not true. Even early on, most mathematicians doubted he had a proof. But I'd check out the history of the problem as described on Wikipedia. There was considerable research into the problem before Fermat, and general research into Diophantine equations had gone on for over a thousand years before Fermat. I have no doubt there would have been significant interest into solving the problem even if Fermat had never written his "I have a proof but it's too big to fit in the margin" note.