Game Center keeps looping on my iPad. Any fix besides wiping it?

If several games are affected on one iPad while they work on another device, that points more toward the iPad’s Game Center or system software than one game. Since the fixes only last a few minutes, I wouldn’t keep repeating resets or erase the whole device without a backup and a clear reason.

Before doing anything drastic, check that the iPad’s date and time are set automatically, restart it, and make sure iPadOS and the affected games are up to date. Then test one game. If the loop returns, note the software version and report it through Apple Support, since a temporary improvement after resetting settings doesn’t establish a lasting fix.

A full erase might seem like the next step, but reports here suggest it may only clear the problem temporarily. I’d avoid that unless you have a current backup and support recommends it. If this started after a recent update, waiting for a system update or contacting Apple is a more sensible next move than repeatedly signing out of your Apple Account.