On my mac book pro M1, I have a bluetooth Keyboard Nuphy Air75 v2, a bluetooh Logitech MX Anywhare 3, and Airpod Max.
After some more testing, I found that the Airpod Max had conflict with other bluetooth Headsets, any brand not just Apple.
In my case, it is clearly that the Logitech's mouse MX anywhere 3 or other Logitech mouse caused the conflict with bluetooth headset, any brand, not just apple brand.
As soon I turned off the power from the mouse, the sound on headset came back clear like before.
So I decided to connect the mouse using the Logitech software call Logitech Option+ via its USB wireless receiver for now.
I don't know if this is the Logitech's driver issue or Apple's issue.
Hope someone from apple give us some more information on this.