At the moment, my PS5 controller connected to my Linux PC via USB-C.
It has perfect support due to official in kernel drivers from Sony and very little lag when used via USB-C.
I'm switching to Linux when I get set up again. Fuck the Recall bull shit. Straw that broke the camels back. I had not considered controller drivers. Is Xbox One over Bluetooth a thing, or will I need to go with Sony?
The XBOX One controller should be fully bluetooth hid compliant and it should work out of the box for at least all the buttons and axis.
There are userspace and kernel drivers for the XBOX controller too (xpad and xboxdrv) but I don't have much experience with them or with bluetooth controller in general.