I am using pdp Xbox One wired controller for PC without any problems on steam link on my RasPi. Well, except, that the cable is almost too short :D

Ok cool. I'm having that same problem with the cable being just a tad shorter than I'd like. I think I'll spring for a longer cable instead of a new controller or adapter.

You'll get the best results connecting it via VirtualHere, if you're using a wireless adapter (or even wired for some controllers). Don't mess around with trying to get it to work natively in Linux and be detected as a controller by steam link, because that'll cause issues with the joystick curves being wrong (among other things)