Mega Man X on the SNES is an amazing trilogy and still holds up. Moving on, X4 and X5 are great to play as well, X6 and beyond can be avoided.
Mega Man Legends 1 and 2 (PSX) are interesting games, a mix of action platforming and light RPG elements, but I have deep nostalgia and still enjoy the games, so you may find them very archaic. Playing the first game, the first thing you should do is change controls so turning around is left/right and L1/R1 strafe.
If you haven’t tried them yet, Donkey Kong Country 1, 2 and 3 are well worth playing, but the games can be unforgiving, especially in later levels. While there’s no coyote time, doing a forward attack off a ledge will allow you to jump once anytime during the fall. It’s very clearly a deliberate feature, as some level skips can only be accessed with that trick.
Castlevania Symphony of the Night on the PSX, if you haven’t played yet, you definitely should. This motherfucker stood the test of time with gold marks.
I know you asked to avoid final fantasy-esque titles, but Final Fantasy Tactics (PSX) is worth checking out. If the battle system, or the party management/job system, don’t pique your interest (things that you should get a feel within the first 20-30 minutes), then it’s not for you, no problem.
Tekken 3 on the PSX is probably the best 3D fighting game on the console. Graphically ancient by today’s standards, but it still has its charm. Later games got better with more characters and everything, but that game has the normal arcade mode, a “volley” mode and a beat-em’ up mode.
Bomberman games are their own category. The SNES ones are some of the best, but playing them all in sequence will get really tiresome, as the formula doesn’t change. Bomberman World on PSX can be completely avoided, it sucks.