Be aware that maybe I am not very updated, Odin Pro was the best thing ever that is not the Steam Deck last time I checked and this market moves fast (albeit it doesn’t feel like it) so as I commented before, RetroGameCorps is the best source to keep you informed about this amazing niche gaming sector.
Even when I might sound contradictory I still think Odin Pro might be the best choice for PS2 with a tight budget… But it is not perfect, if you want perfection Steam Deck is the way to go.
You could also reuse your own hardware if you happen to have a competent phone, just buy a telescopic controller and then you can try PS2 games in your Android device, not the best in the world, but you might get a better idea of how this is done.
Odin Pro I think it has a SD 845 and runs on Android, so it is basically that, a phone with a controller attached lol.