I got through phases of games, and right now I'm in the one I like to think of everything-modern-is-making-me-angry-so-I'll-turn-back-to-vintage-games - and that's in the form of PCSX2 on my Steam Deck.
Playing the PS2 generation on the Steam Deck is kinda the 'perfect' generation: I can upscale the resolution without a performance hit, I can apply HD texture packs, Retro Achievements and I can drop the TDP to extend the battery life.
Anyway, I never had a PS2 (it was just a touch before my time), so there's so many games I'm yet to discover there. I'd love to hear any suggestions you might have for what I can play on there:
Just a couple notes:
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No GTA games because they're better with the PC versions. While I've bought the games, the account that Rockstar demands people have to play their games makes me angry on principle (a single-player game forcing an online check each week to be able to play it? No thanks), so I've got cracked versions to play.
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No Final Fantasy because I'm not even sure where to start!
So if anyone was around back in the PS2 era, and has some recommendations for games they've loved, I'm all ears. I've been having such a fun time playing these.
Some big thanks to my friends who made and run RetroDECK (my choice for emulating on the Steam Deck), and to the PCSX2 team (who I'm chatting to right now actually, so keep an eye out and in a week or so I'll share that chat with them all about creating and maintaining the PS2 emulator!)
Also, if you do have a Steam Deck, and want to play with the HD texture packs I do, then it's really simple. Just download the pack, and name then according to their game's I.D. code - SLUS-20743 for example, then place it in the texture_packs folder:
From there, go into PCSX2, then right-click the game, select properties, graphics, texture replacement and tick the two boxes I have:
Then you've got the games looking the best they can be!
Here's a link to a stack of pre-done HD textures for a lot of PS2 games, to help you on your way!

Tony Hawk's Pro Skater 3 was pretty close to the peak of the series if you ask me, and the PS2 version was the superior one. THPS4 also came out on the Playstation 2. I see you already have Underground on there.
If you'd like something you can handily use to consume the rest of your entire life, Disgaea and/or its sequel will probably do you.
Ico and Shadow of the Colossus are also legendary. I haven't tried in ages, I have no idea if modern emulators can get the latter to run at a non-crap frame rate. It'd be a lot nicer if so.
Odin Sphere is an often overlooked 2D action sidescrolling fighting thing wherein you Norse In The North and beat the shit out of absolutely everyone. Its sequel, Muramasa: The Demon Blade is much the same thing except therein you Ninja In The Night instead. The latter stayed locked to the Wii to my knowledge but the former was on the PS2.
The PS2's library is quite vast. I'm not going to go looking this up to prove it right now, but I'm pretty sure it's got the most titles ever released for a home video game console (i.e. not the PC) in history. Even just trying out unknown games at complete random, it's likely to be able to keep you entertained in one way or another basically forever.