Many people facing this Shareplay not working error after the latest iOS updates. I think you should wait for some new updates of iOS 15. I hope in the next updates this Shareplay issue gets resolved.

You can easily fix this Game Centre unable to connect to network issue with the help of these steps: Firstly, Go into settings > iCloud - turn off the "Game Center App". Next, turn on this button Then go to settings > Game Center > logged in. I hope this may resolve your Game Center unable to connect issue.

You should follow this thread. https://developer.apple.com/forums/thread/5464