Same problem. Bluetooth turns off automatically and gets in the way of productivity. I don't recall this to be in the list of updates. I had another problem with Bluetooth after upgrading to Monterey. My headphone started beeping twice periodically when I am in meetings and disables its microphone. I did not have this problem with the same headset with BigSur. I tried another identical headset and it shows the same problem. So this is not the problem of the headset. Something changed on the MBP side. I have a 2019 16" MBP, now I am running MacOS 12.3.1