Suddenly unable to open/log into Facetime

I have been using Facetime without problems on all my Macs & iPhones for quite a few years. I made my last FT call 5 days ago, so it was working then. But today ... on all my Macs ... FT wants me to sign in with my AppleID but I am unable to. Every time I enter my Apple ID & p/w there is a pause as if it's signing in but it just returns the same sign-in window:

Apple online help suggested going into Facetime Preferences to sign out/sign in but I cannot do that because the prefs are greyed out. This is the same on all 3 different Macs:

But it works fine on my iPhone 16Pro, which is on the same network as my Macs (using WiFi not cellular).

System Details:

2018 i7 Mac mini - Sonoma 14.7.4 - Facetime 5.0

2019 i9 iMac - Monterey 12.7.6 - Facetime 5.0

2025 M4Pro MacBook Pro - Sequoia 15.4 - Facetime 36

2025 iPhone 16Pro - iOS 18.6.2 - Facetime version unknown

Testing

Macs - Shut down & restarted each Mac; also tried different user accounts on each Mac. Same FT behavior every time.

Network - Shut down & restarted my entire network infrastructure ... FIOS NID, Router, Ethernet switch, WiFi AP. No change in Facetime behavior on my Macs.

iPhone - Facetime works perfectly fine on my iPhone on the same network my Macs are on.

AppleID - Checked settings in each Mac, iPhone and Apple Account Support.

Everything except Facetime is working normally.

I also scoured the Facetime Community here, as well as generally on the internet. There have been similar reports from other users over the past few years but no apparent consistency or solutions.

At this point I am inclined to think the problem lies somewhere at Apple ... but Apple Service Status indicates Facetime is up, and in fact it must be up because I can use my iPhone to make & receive FT calls!

Reaching out for any suggestions about what to look into next.