I tried the new admin user, safe mode, and the 'force quit' of imagent fixes, and the problem persisted in all three scenarios. No luck; frustration builds.
I have a suspicion that this is somehow linked, somehow, to AppleID -- I bought this MacBook Pro in July 2020, having lived on a 2009 MacBook for a decade. I don't own a smartphone of any kind, and am not savvy with any of this stuff. We also bought an AppleTV at the same time. When I set this new one up, and set up iCloud, and seemed to configure everything pretty smoothly on my AppleID, everything worked brilliantly -- except a few days later there appeared a software update for Pages, Numbers, and Keynote, and when I tried to activate the update, it said that my AppleID wasn't the right one linked with those updates, which I thought very strange, but because I had no use for those apps, I just carried on. Those [3] updates just sat there for months, then 2 more were added in with Garage Band and iMovie, leaving [5] un-updatable for a long period of time, but again, I never used them so whatever.
On the same morning that iMessages and FaceTime stopped working, as I followed instructions to update my computer with the latest, those 5 updates suddenly worked -- voila, and updated accordingly -- but now iMessage and FaceTime do not. That seems too coincidental to be a coincidence, to me, but I don't know much about this stuff. The constant request from these two apps to re-enter my AppleID only to shut me back out and start all over again seems, again, like there is something rotten in the state of AppleIDdom.
Shall I contact Apple Support and spend some time with someone (i.e., can they press a magic button that will bring inner harmony and peace to my AppleID and, by extension, my computer)?