I spoke to soon. This morning, the hands-free speaker is no longer visible in the MS Teams devices pane. And, like before, if I try and use the other WH-1000XM4 headset profile, I lose sound when choosing another speaker.
Summary:
* Friday: application of the 22H2 update restored the hands-free headset
* Monday: hands-free was no longer present after starting up the laptop
So, it looks like the operating system knows about the hands-free bluetooth profile, but something (teams?) is disabling it.