Hey rabarbera,
It appears that the microphone on your Bluetooth headphones stop working during Zoom meetings and Google Hangouts when using them with your mac on macOS Monterey. Good job in making sure your Mac is updated as well as testing a different browser, and we're glad to provide assistance.
Change Privacy preferences on Mac - Apple Support -- Go to the Apple menu () > System Preferences > Privacy > Microphone. Are both Zoom and Google Hangouts being given permission to use your Mac's microphone? With your Bluetooth headphones selected as the input, your Mac should allow your headset's microphones in lieu of the built in microphone on your Mac.
How to use safe mode on your Mac - Apple Support -- If the built-in microphone has been granted permission, see if the issue happens when you restart your Mac in safe mode. Whether it does or not, restart your Mac normally to exit safe mode. If your headset's microphone worked while in safe mode, see if it works now.
Please be aware: "When you start up in safe mode, your Mac prevents some software, such as startup items, from loading, and it performs a check of your startup disk. Your Mac may take longer to start up because of the check."
New user or group setup - Apple Support -- If it didn't work in safe mode or stopped working after restarting normally, create a new Administrator account. Log in to the new account you just created and test the issue there.
Please let us know the results when you're able by replying to this post.
All the best.