Despite Fermat's brilliance, it's barely feasible that such a proof exists after 300 years of attempts by professionals and amateurs alike.
Ironically though, the tantalising idea of the existence of a simple proof - motivated by that infamous quote - contributed massively (along with the simplicity of the conjecture itself) to the theorem's notoriety and the myriad attempts to solve it.
He wrote the note to himself, a long time before he died. He then publicly announced a proof for the n=4 case, and nothing about the general case. It strongly suggests that he realized his proof was wrong.
It seems there's a fair chance it was a flawed proof, in which case I doubt we'll ever be particularly confident although we could possibly surface some candidates.
Something I always wondered is if we know of the existence of a proof that is both simple and also non-obviously flawed and as such could have been Fermat's solution?
Ironically though, the tantalising idea of the existence of a simple proof - motivated by that infamous quote - contributed massively (along with the simplicity of the conjecture itself) to the theorem's notoriety and the myriad attempts to solve it.