I just figured this out for my own systems, which have been slowly seeing Bluetooth headsets stop working in this manner for weeks (actually felt like it was "spreading" from one computer to another)
TL;DR - Your antivirus or EDR may be blocking the RFCOMM Protocol TDI device from activating or loading, which will kill headset use.
Sequence of events
1 - Bluetooth working fine for calls and media
2 - Windows update (not the actual culprit, but the reboot after)
3 - Bluetooth headset option no longer available in Teams
3a - Bluetooth device only listed as "connected audio" and not "connected mic, audio"
3b - Headset device shows "disconnected" in advanced audio settings
Bluetooth driver updates, headset driver updates, all do not fix anything. Uninstalling in Device Manager and reinstalling devices does not help.
In Device Manager, noticed an issue with the "Bluetooth Device (RFCOMM Protocol TDI)" device - the driver was "blocked from loading" or the device was disabled.
Looking at the device's drivers and attempting to rollback/reinstall, noticed a third driver listed in the Drivers tab: "C:\Windows\System32\Drivers\SentinelOne\23.2.3.358\SentinelDeviceControl.sys"
SentinelOne has a "device control" module, and one of the default options in the console is now "Bluetooth: Disable RFCOMM for all devices (Windows only)" - this was not present before. Disabling this option in SentinelOne's device control console, and rebooting the computer, fixed my issue. I tested further afterward by disabling the RFCOMM device in Device Manager, and immediately the microphone option was gone from Teams. Re-enabling the RFCOMM device caused the microphone to come back right away.
RFCOMM can be used to transfer files to a Bluetooth device and may be disabled as part of a DLP policy. However, RFCOMM is also how Windows tells the headset to switch to microphone mode - the audio quality drops because there is not enough bandwidth to support both high-quality stereo sound AND the microphone. So Windows sends a message via RFCOMM to the headset to turn on the mic. Without RFCOMM available, Windows can't do that, and can't turn on the microphone.