Game Center unavailable in iOS 13.2.3
I am having the same issue that I found another user previously has in this thread: https://discussions.apple.com/thread/250790235
No app is able to connect to Game Center. Any apps that use it to log in simply freeze or fail, and trying to open Game Center from the Settings app freezes the Settings. This has already cost me money in non-functional games.
The approved solution in the other thread did not work for me: a network reset and logging in and out of Apple ID did not solve my issue. It is not likely network-related.
One game that used Game Center for achievements displayed the attached error when trying to load the achievement screen.
iPhone XR, iOS 13