Apple Arcade games not logging into correct Game Center account on apple tv

We have an Apple TV running TVOS 17.2. My 7-year old kids have an Apple ID, and they have a user account on the Apple TV. They have their own Game Center account. When we switch users on the Apple TV to their account and then open an Apple Arcade game, usually the first game we open will log into their Game Center account correctly but any subsequent games revert to my Game Center account. I am the default user, in case it matters. Is there a way to manually switch Game Center accounts in Apple Arcade? I haven't been able to find one. Or is there a setting somewhere I should change?

Thanks!