I recently tried. After hours of trying my xbox controller still can't connect. Steam doesn't detect my Gullikit controller and frankly, I have to literally decide to go down this pain again to hopefully maybe switch one day.
That's really weird, I also have a Gullikit controller (the one with the hall effect sticks) and it works perfect OOTB on my desktop & laptop, both running Linux. Maybe try switching the mode on the controller to Windows/Android instead of Nintendo Switch?
Don't have one of those (and really don't want one, AA ftw in a controller tbh), from what I read during my journey is that the xbox controller uses some weird DRM in their chipset but I have honestly no clue what that means in the context of bluetooth.
Odd. Are you sure it's Linux? Or at least totally Linux? I've had no issues pairing MS Series X controllers (or Xbox One) over Bluetooth to my Steam Deck or Linux PCs or using them plugged in. Maybe the Gullikit controllers don't operate quite the same as the MS versions that the drivers don't play nice?
Honestly I connected it to both macOS and windows before. I recently got a new WiFi card so I'm thinking it just has a crappy chipset in terms of Linux support (Asus PCE-AXE59BT is all my store had at the time). Playing around with Mint and tried the xpadneo kmod.