Linux Gaming

In my experience yes they do. Having used Ubuntu, Vanilla OS, Manjaro, and now Fedora Kinoite on my desktop rig, bluetooth controllers like the Xbox One and Switch Pro work exactly as you would expect, no idea about wired controllers though.

Of course it's going to be up to the specific game as to whether it has controller support or not, but adding the non-steam game to Steam would give you that along with controller remapping if the game doesn't natively support it.