Hi,
I have the exact same problem here: I have the Bose NC 700 and in Windows 10 it allowed me to select the hands-free "device" for my voice over IP phone app while the high quality audio (streaming) would be played using the other (16bit) Bose NC 700 "device" for high quality audio.
Disabling the hands-free mode to get good audio (listening) quality is not an opiton: I use the hands-free for my work-calls VOIP and at the same time, I listen to music using the same headset (when a call comes in, the music is muted automatically in windows 10 during the call).
Now, after upgrading to windows 11, there seems to be no known way to get both modes working as it was in Windows 10.
If Microsoft wants to hide the complexity from the user, that is fine by me. I have to say that it confused me to see 2 devices and that the sound quality was bad on one and I had to google etc.
But if Microsoft chooses to hide the complexity, It must make sure that audio streaming goes to the high quality audio device and the low quality hands-free mode is only used for calls.
Interestingly, in System -> Sound -> Properties, the Recording device (that is the hands free one, I guess) in the settings of Windows 11, you can choose if it should be the standard for communication or the "standard for audio" (translated from German, as I use Windows 11 in German).
As the hands-free one after installing the headset is only set to "standard for communication", I would assume that in the sound settings, the Microsoft developers intended to make a distinction to use a devise for communication = hands-free or as an audio-device=playback of streams. However, somehow, some other parts of Windows 11 do not seem to respect this setting (if my interpretation of the sense of this setting is correct).
Long story short: microsoft needs to fix it!