OpenTyrian and OpenTTD were my go-to games for many years. Both are easily found for Mint packaged with free assets.

Minetest, Mindustry, 0AD. I can’t speak for how well they’d run on Core 2 but I think they’d be fine.

OpenRA will get you some classic Command and Conquer. You can run them as Appimages on Mint, and on launch they give they option to download freely available assets.

FreeCiv if you want to get lost in some old school Civilisation.

GZdoom you can play with the FreeDoom assets.

And then there are some open source engines but you need to pair with commercial assets: CorsixTH for Theme Hospital ScummVM can run a whole bunch of older adventure games (some might even be free if you have a gog account) OpenRCT2 for Rollercoaster Tycoon 2 DevilutionX for og Diablo

Besides the already named ones, here a few:

• endless sky: Inspired by a game called escape velocity. You fly around in a spaceship and do space adventures.
• naev: Similar to above, but quite different.
• ddnet: Multiplayer coop 2D platforming. (Has enough active players.)
• warzone 2100: Real time strategy with tanks. Has a campaign.
• battle for wesnoth: Medieval turn based strategy game.
• mari0: 2D mario with protal gun.
• mindustry: Build factories and defend against enemies.

Lately I'm having a good time playing Sonic Robo Blast 2, basically an "old school style Sonic game that uses Doom's engine to make it 3D". It's a very polished experience and should run fine on anything.
Another long time favourite of mine is Shattered Pixel Dungeon, a roguelike dungeon crawler that has active development and a great community

https://www.openhv.net/ is a cool project especially for multiplayer.

https://www.minetest.net/ will probably also run and has loads of cool game mods, not just the Mineclone2 one that is very similar to Minecraft.

GZDoom!! There are so many great mods! Search YouTube for some "best" mod recommendations.

I'm a big fan of cataclysm:dda. It's a rougelike, so it needs some time to get good at, but there is a ton of content and I can run on a potato. It's got violence against zombies, but with even the most advanced graphics it can use, it isn't particularly graphic.

Minetest, and in particular MineClone2. Always someone playing multiplayer also which is quite nice.

https://git.minetest.land/MineClone2/MineClone2/src/branch/master/HOW_TO_PLAY.md

Mindustry is good fun also :)

https://veloren.net/ - I couldn't find system requirements but maybe that will help to run it.

https://libregamewiki.org/Main_Page

Maybe some older games that don't require downloads: https://archive.org/details/internetarcade

Gotta recommend WesNoth https://www.wesnoth.org/

While the nostalgia part is likely lost one someone that didn't grow up with this style of game It's still an excellent strategy game with an almost puzzle like feel to it. Bonus points for being in development for longer than you've been alive as well I feel ;)

Old school adventure games are great for these sorts of use cases, though they may not be to your taste since they can be a little dense. Grab yourself Frotz to gain access to the entire back catalog of Infocom's text adventures plus a entire library of modern user-made selections.

ScummVM also gets you access to a decade or two of LucasArts and Sierra Online's classics.

Rimworld is only as explicit as you make it. I mean I guess people try to kill your people, so there's blood, but non of the war crimes are requirements just emergent gameplay features.

I like Endless Sky - a free and open source space trading, combat and adventure game.

Rogue and Roguelikes...

DoomRL AlienRL DiabloRL Rogue

factorio and stardew valley.

games like pubg csgo and fortnite might run if you have a video card and 8gb ram.

hell, many AAAs from the ps3 era will run fine depending on your video card.

The UrQuan Masters: https://sc2.sourceforge.net/ or https://store.steampowered.com/app/2645580/Free_Stars_The_UrQuan_Masters/

Even if you're not running Linux, check out the archwiki:

https://wiki.archlinux.org/title/List_of_games

These will run on pretty much anything, and many of them are fantastic. Check this out!

So you're saying you want free games with explicit content? I'm sure Bible Black would run well on it.

(Yes, I'm kidding.)

Xonotic.

Great arena shooter all around even excluding the fact that's FOSS. Used to play it with a 2012 Intel i5 and HD 2500 integrated graphics on 1080p medium settings and was averaging around 80-90 FPS on default maps.

Now I don't know what kind of GPU do you have, but I bet that it would probably just fine on low settings and maybe at a lower resolution.

Another thing worth mentioning is that you may have trouble finding populated games with good ping if you live outside of Europe.

Noita would probably run on that. Amazing fun game

Super tux kart Battle for wesnoth Shattered pixel dungeon

FTL

I personally really like osu! and it's available as a flatpak for linux (you should be able to just install it through the software install thingy)