2000s are quite "recent". It sounds like you are more into Flash like web games and apps? I haven't really gone through that era, so it's hard to say. I don't know what's good and if someone persisted it. From this time I remember things like https://www.newgrounds.com/ , which I think has games too...

If you would be interested in the era before that, then there's a very large community for retrogaming - essentially all games from things like PlayStation, GameCube, etc can be downloaded and run in an emulator. Either on your computer, or there is a very large variety of retrogaming devices that cost somewhere between $50 and $300 (https://www.youtube.com/c/RetroGameCorps is probably the best channel).

Another option that gives you that vibe would be to explore pico8 games https://www.lexaloffle.com/bbs/?cat=7&carts_tab=1#mode=carts&sub=2 . Check out my list from #388261