If, like me, any of you saw all these posters saying "that worked!" in reference to all manner of crazy idea, only to painstakingly try them all and no, it still didn't work, you might try something I found a different thread where someone did this:
"Log out of FaceTime and Messages on your Mac, and then log back in."
This worked for me. Which I actually find very odd. I also find it odd that there are so many weirdly different solutions.
This leads me to believe that one or two of these other ideas are the actual fix, but that fix is followed by a time delay (communicating with Apple ID or some other data propagation between devices). Since we do not see an immediate result, we try all sorts of other unrelated things, and then suddenly, lo and behold, some totally unrelated action appears to work only because sufficient time passed since we did the thing that ACTUALLY worked.
Anyway, could be wrong, but there are so many different supposed solutions listed across so many threads that I doubt the veracity of many of them - including the one that appears to have worked for me.
Very annoying problem, no matter!