On my RG300X I also follow a rotation system so I kind of see what you're coming from and why games need to be categorized ("long committed playthrough" vs "quick fun").
Let me start with typical fantasy RPGs.
No action-RPG list on PS1 is complete without Threads of Fate and considering everything you've said this might be a homerun. It's a straight action game with limited character progression, it has great level design with lots of platforming and common sense (read: not tedious) puzzles, the 3D is beautiful and the OST has that typical 90s Japan feel of wanderlust and wonderment. The plot is not too bad either, the two characters have their own arc and you'll need to play each to reach the final ending.
Another action-RPG you shouldn't miss is Legend of Mana. I really can't put into words how impossibly beautiful it is, especially on a small screen.
Tales of Eternia (released for some reason as Tales of Destiny II in the West) is another great action-RPG, though the battles don't happen on the overland map but in their own separate screen. In true Tales fashion it provides a great sense of adventure, with beautiful locales and a fitting musical score.
I've never been a fan of Japanese RPGs in my youth, but over time I warmed up to them and now sort of respect how effective they are at communicating emotions, especially with music.
Now for firearms.
Resident Evil 1 leans a bit too much on puzzles and ammo management to qualify as an atmospheric "action" game, but by Resident Evil 2 all of that was already gone: it's squarely an action game with tons of ammo and powerful weapons, and Resident Evil 3 followed suite.
Same thing happened in Dino Crisis: in 1 you have a slow-ish and honestly pretty tedious attempt at a low-resource puzzle game against dinosaurs, and Dino Crisis 2 went full Rambo with a fast and slick gameplay and tons of powerful guns that feel super satisfying to shoot. The graphics are superb and easily among the most beautiful atmospheric environment on PS1, excellent balance of high action and cooldown moments.
Another action/exploration game that often gets overlooked: Chase The Express (titled Cover Ops: Nuclear Dawn in America). Same deal expect in a basic Clancy-like setting.
You're probably already familiar with Parasite Eve 1 and 2.
All of these aren't RPGs by any stretch of the imagination but on PS1 they do a great job of scratching that itch where you gradually explore and backtrack (instead of moving from level to level in a linear game) and acquire gradually more powerful gear.
Some other vaguely similar games include Alone in the Dark The New Nightmare, Fear Effect 1 and 2, Galerians and Martian Gothic Unification, but all of them are more like RE1 than RE2, i.e. you play them if you're looking for a chill puzzle/exploration experience, not action.
Now for quick games.
If you rely on save states then of course any game can qualify as a quick in-and-out game. After all you can quickly hop into say Medal of Honor, play half a level and call it a day.
But some genres are better for that purpose, and you probably know the main ones already:
sports (football, tennis, BMX, skateboard)
racing (Colin McRae, Gran Turismo, Need for Speed)
1v1 combat (Tekken, Mortal Kombat, Soul Edge/Blade, Dead or Alive, Street Fighter)
By the way, if you like driving games without them being strictly racing games, then you have some story-heavy ones like Driver 1 and 2, World's Scariest Police Chases and The Italian Job.
Racing Lagoon is also a pretty unique take on racing as it's intended to be an RPG with cars, but I haven't played it yet.
When it comes to strategy I'm having a hard time putting down Red Alert. The PS1 controls obviously aren't as good as on PC but it's hard to pass on considering it's one of the few real-time base-building games on the platform. You get used to it over time, and needless to say I'm not playing the campaigns but only skirmishes, which puts it into the "quick session" category of my rotation.
If you're interested in turn-based then Nectaris Military Madness has great gritty military art. If you liked Advance Wars then you can view it as a distant cousin.
And lastly, if you like flight games then you can play the Ace Combat games, in particular the English-translated version of Ace Combat 3 Electrosphere as it has a real story that was removed in the Western release. Ace Combat 1 and 2 are of course excellent as quick session games.
Else you have G-Police 1 and 2, the Colony Wars trilogy and Wing Commander III and IV. They have their difference but they converge in that they have a very present story with tons of cutscenes and voiced dialogues despite being linear. To me that's a great mix to get you invested in a "long" game without the pressure or headaches that come with exploration or puzzles.
And that's it. Sorry that I focused on PS1 but this is already a lot longer than I expected, I better leave here without a real high-IQ conclusion. Hopefully this fits what you wanted.