Game Center login issues

Iphone 6s with current iOS (11.1). As far as I know, the only game I play that still uses Game Center is EA Madden Mobile, and I have never had an issue logging in until yesterday, 11/1. The game simply told me it couldn't connect. When I went into the phone Settings - Game Center, everything looked fine. But I turned game center off, and when I tried to turn it back on, it told me this:

I have been able to login successfully 3 times out of probably 100 tries. I have tried shutting down, doing a hard reset, changing my apple id password, and finally even a full reset / restore, but it still behaves like this. Any other ideas out there?