The documentation says "In your game, you need to authenticate the local player before you can use any GameKit APIs and Game Center services".
https://developer.apple.com/documentation/gamekit/authenticating_a_player?language=objc
Game Center Programing Guide says "Saved games are tied to the iCloud account, not the Game Center account" and "Most Game Center classes function only if there is an authenticated player, and those classes implicitly reference the local player."
But I test it on my game, it seems that the game save APIs just works even without Game Center enabled.
I'm not sure if this make scence to all iOS versions.
We only use Game Center for cloud save. If authenticate is not necessary, we prefer to remove game center authenticate because it may cause deadlock through Unity API.
https://forum.unity.com/threads/ios-authentication-issue-social-ilocaluser-authenticate-caused-user-deadlock.975390/