For the older 360 there was, not paid much attention to what goes for the xbone, or slim/lite variations of the 360. Obviously the original xbox is wired, and its USB adapter consists of "buy USB extension cable, get xbox breakaway controller bit, solder like coloured wires to like coloured wires leaving the yellow intact" (leave the USB female if you want as you can use a select handful of small and probably hard to come by USB drives on your homebrew xbox in that case to transfer files around at USB1.1 speed -- basically just use it for save games and songs).
"Maybe it's possible to hack something but I would just check which controller you have a perhaps look at the official PC adapter or BT."
The official adapters are both rare as rocking horse shit and generally considered awful (presumably why they did not stick around). They also did not use bluetooth as much as a proprietary radio signal which is why we are in this dilemma.
Personally I wire everything I can but
https://www.instructables.com/DIY-Xbox-Controller-Receiver-for-PC/ seems reasonable if you have some old RRoD things you want to make some use of.
https://techocd.blogspot.com/2014/01/xbox-360-wireless-controller-to-pc-via.html is a bit older and more complicated but has some choice info.
