I have the same issue described, and have found a workaround.
O/S: Monterey 12.2.1 (issued surfaced on first upgrade to Monterey)
Headset: Aeropex Aftershokz
Application: Google Chrome 98.0.4758.102 (all VC apps: Google Meet, Zoom, etc)
Fix: In the VC app, open sound settings. Change the microphone setting to something besides the headset. In my case, I keep output through the Aeropex, and use my webcam for the microphone. It stops the forced muting of the headset and the loud beeping sound.
Issue: I suspect some 3rd party headsets are using the wrong connection profile -- connecting as a stereo device, rather than a communications device. The muting only happens on the first instance of speaking, so it appears to be triggered when there is a microphone signal. This leads to the headset being sent the signal to shut down the input channel. I cannot find a setting in MacOS to force the connection profile for a headset, but my guess is that Monterey introduced a bug/feature that tries to correct for unexpected input.