I'll list off some of the older titles on my faves list

• Earthbound is on there, so glad you got that one :)
• Mother 3 (GBA)
• Mario and Luigi superstar saga (GBA)
• Super Mario World (SNES)
• Super Metroid, zero mission, fusion (SNES and GBA)
• Myst (lots of platforms these days but originally pc)
• Resident evil 1-4 (ps1-2)
• Kirby superstar (SNES) / Kirby superstar ultra (DS)
• Silent Hill 1-4 (ps1-2)

I'm def missing some here. I would argue not all of these have necessarily stood the test of time if we look at it from a modern sensibility lens. Particularly resident evil 1-3 and silent hill 1 can be a bit difficult to play if you can't get used to tank controls. And Myst can be a bit esoteric, especially the older versions.

Besides that though I really do think these have all stood the test of time. I would like to particularly shout out the original re4. Despite having a recent remake that I adore potentially more than the original, the original is still so good even to this day. It is wild to me how well it still feels to play

Arcade:

• Capcom vs. SNK 2: The Groove system is one of the coolest dynamics to tailor the game to your playstyle. Is it balanced? Hell no, but I love this game casually.
• The King of Fighters 2002: KOF fans will tell you either 98 or 02 were the absolute pinnacle. I side with 02 because it has Kula in it. Also note that 98 and 02 both have updated rereleases with an extended roster and rebalancing, but those are Windows-only.
• Puzzle Bobble 1/3: You've probably played some flash game clone of this. IMO I think 1 was best for its simplicity, I'm not as fond of the garbage patterns introduced in later titles in an effort to give characters some asymmetry. But PB1 does not have AI opponents, singleplayer is only the stage clear mode, so if you don't have a human to play with try PB3 for the next best thing.
• Soldam: The singleplayer modes are nothing to write home about, but it has one of the most unique versus modes I've seen in a puzzler. Shared piece queues are normally horrifying, but Soldam makes it work by giving P1 the objective to match red while P2 matches blue. So if you want to snipe pieces that are desirable to your opponent, that means taking pieces undesirable to yourself. Garbage is also based on how you clear lines, so crafting maximally disruptive garbage gets interesting. The catch, unfortunately, is that there is no AI. But if you can play this game with a human, do check it out.
• Tetris: The Grand Master 1/2/3: The only good Tetris, do not @ me. Start with TGM2's Novice Mode, then once you can clear that go back to TGM1.
• Twinkle Star Sprites: A versus shmup with a very unique format. Chaining enemies on your screen sends attacks to your opponent's screen. Hard to really explain, just give this a spin and feel it out for yourself. There are a lot of moving parts, screenwatching is vital, and feels like I've barely scratched the surface of the game's depth.
• Vampire Savior: Aka Darkstalkers 3. This game is fast as hell and it's a blast. Like with any classic fighter, good luck keeping up with FightCade folks who really know what they're doing, but I love it casually.
• Waku Waku 7: This game's mechanics are honestly borderline kusoge, you can't even cancel normals into specials. But I love the design and atmosphere so much. Tesse is really fun to play even in spite of the system mechanics.

NES:

• Fire 'n Ice: A very rad little puzzle game.
• Mike Tyson's Punch-Out!!: Just an absolute blast. I won't bother listing them seperately but also check out Super and Wii. Super's kinda the black sheep of the series, but it's still a good game, just not as good. Wii is an absolutely top-notch successor and I'm sad it didn't get any more sequels after that. The two arcade predecessors are honestly forgettable.

SNES:

• Chrono Trigger: I am hesitant to recommend most JRPGs from this era if you did not grow up on them, because many of them haven't aged so gracefully. Chrono Trigger is the exception, this game is a fine wine. You may want to check out one of the rereleases though, or at least a retranslation patch, because the original translation was made on a rushed deadline and bound by heavy technical limitations.
• Earthbound: A bit more of a slow burn in comparison to CT, but this game is carried by incredible writing. It's also required reading before playing Mother 3 next.
• Kirby Super Star: Definitely the peak of the series, giving every copy power an entire moveset is a blast. Has an updated rerelease on DS with added extras, I do highly recommend this version, but DS can be awkward to emulate so SNES is fine.
• Wario's Woods: The NES version is more well known since it was the system's last first-party title, and for whatever reason it's the only version Nintendo ever rereleases. But the SNES version is a notable upgrade, biggest thing it has is AI to play versus mode against. Versus mode is wild as hell, so if you've never seen it please check out the SNES version.

N64:

• Dr. Mario 64: Best version, but can be notoriously difficult to emulate. If you have issues with it, SNES is a good alternative. Don't play NES.
• Mario Party 2: Still the best in the series.
• Paper Mario: Pure perfection. Many fans will say TTYD was better, and it's certainly a good game too, but I think 64 was peak simply because the pacing is so much better.
• Super Mario 64: It's Super Mario 64. You do not need me to tell you that this game is good.

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.

Thief: The Dark Project and Thief 2: The Metal Age (PC, 1998/2000). (the second game is essentially just more of the first, but more refined)

Still the best stealth games to date if that's your cup of tea, and holds up excellently.

• Kirby's Adventure (NES)
• Final Fantasy Tactics Advance (GBA)
• Chrono Trigger (SNES)
• Ninja Turtles 3: The Manhattan Project (NES)
• Battletoads (NES)
• Little Nemo: Dreammaster (NES)
• Mega Man 2 (NES)
• Mario 64 (N64)
• Starfox 64 (N64)

I know you want to avoid Final Fantasy style games, but both Chrono Trigger and FF Tactics Advance are unique takes on JRPGs (you're open to Earthbound, after all). They are also timeless, IMO, and stand up against modern titles.

Kirby's Adventure is great all around. Has saves, interesting gameplay, tight controls, excellent music, secondary secret objectives, and it's just goddamn adorable. It's also better than Kirby Superstar, IMO, but do try both.

Oh, and just in the moment I hit send, I remembered another gem from the olden times:

Unreal World: Basically the survival game. 99% of today's survival games are just a pale shadow of this. I mean, nowadays there are even "survival" games without hunger mechanics or proper simulation of wounds... No, this is not one of those easy mode survival games. This is Fantasy Finland, and it's the Fantasy Iron Age. Available for free or, if you want updates faster, also for money on Steam.

If you have any interest at all in Souls-likes and want to see where the genre really started, consider Blade of Darkness. An updated version is available on GOG with modern resolution support and slightly improved controls (still clunky as hell in comparison to newer games, though).

You have tons of great game suggestions here so i'm not gonna suggest another game. I'm just going to recommend RetroAchievements as a way to add a little meta-progression fun to your retro journey, if you aren't using them already. They integrate into various popular emulators (RetroArch especially), and it's fun to be playing Super Mario World and get a little extra recognition for beating a boss.

Don't know if it'd be something worth while, but if you're looking for a game to just pass the time, Devil Dice (or XI in Japan I think) for PSx. Trailer linked is highest quality I could find.

If I weren't currently at work and would have time to think about the answer, I could probably come up with more titles, but those are the top 2 that come to mind, if I ignore cRPGs (at least that's how I read your "avoiding final fantasy-esque" requirement):

Settlers 2: It's new enough to still look decent by today's standards, and has amazing game design. Available at GoG.

Star Control 2: One of the best early open world games. The graphics have definitely aged by today's standard, but the humour hasn't. Or maybe it has, but just a bit. Available for free and open source.

Ace Combat (PS2): Primarily Ace Combat 4, I’d say as it’s shorter but still great. If you like the gameplay then you’ll need to play 5 and Zero.

Metal Gear Solid is one of my favorite games of all time, followed closely by 2 and 3. It really is an incredibly timeless piece of art in my opinion, just filled to the brim with incredible moments.