This started for me yesterday. None of my three Macs can use either Messages or FaceTime. They both close and return to the login screen, as you say. Coincidentally (?), I had put Tahoe RC on my M1 the day before. It had been running Tahoe betas all summer. OCLP is not an issue in my case. The other two Macs are Intel, on 15.6.1.
Jerry, my solution was to speak with a senior advisor at Apple support and they found (after a couple of days) that my iMessages and FaceTime account had been blocked for some reason but wouldn't say why. They took me to a special website where I was able to request an unblock and it all came good 12 hours later. Usually, the only reason Apple blocks is for spamming, but I have never used spam. I'm not a great believer in coincidence, so I'm fairly sure it's related to the Tahoe RC install but we'll know more when millions of people install the release version on Monday.
Well as I've said before, if the storage in this iMac fails, it's not worth it for me to fix, so I'll toss it and get a new computer, and get all new hardware.