Same problem here.
Bose QC35 II well connected by BT, recognised as a headset everywhere but not listed as audio input/output device.
It was working perfectly on Windows10, worked when upgraded to 11 but suddenly stopped with no reason.
I cannot choose it as output device because is not listed as a capable device.
I've tried all those newbie solutions that MS copy&paste in every post related to this issue... it's sad to read that years later the issue stills there.