I played gay dodgeball years ago and had a ton of fun with it. Been playing a lot of halo infinite lately and wishing I could do so in a rec league type environment.
A cursory internet search didn't turn up what I'm looking for. Do these exist? Should they?
This would vary massively from game to game. There are plenty of amateur/for fun leagues which accept "average" players for League of Legends, for example. But yeah, as someone else said, you kinda need to be looking for Discord servers. Find any discord server related to the game you wanna play and ask around, you'll probably get pointed to a different discord server if there is such a thing. Discord's where everything game related takes place these days.