Thank you to CS-17.
As they mentioned, turning Game Center OFF under APPS USING ICLOUD, and then back ON, sort of fixed it. However, that actually just replaced the previous Apple ID for Game Center with the main Apple ID for iCloud. Had to then sign out of that ID for Game Center, and then back in with the correct Apple ID from before iOS 14 dropped it. And upon doing that, just ended up back with the same network connectivity error as before.
So, the bug is either associated with the use of different Apple IDs for iCloud vs. Game Center, and/or with having that different ID being not an email address (old school).
Still waiting for a solution here.