I run Bazzite and I have a few 8bitdo controllers. I've only ever used them via bluetooth, but once they are paired they work perfectly.
I do have a Thunderobot G45 I use via dongle. It plugs in and works, no configuration or software needed. And after I set up my system settings properly, I can also use it to wake the system from sleep. But that was a Linux thing, had nothing to do with the controller.
Any of the major distros will probably work with any big-name game controller.