Many people have had the same problem as you. I think you are also supposed to enable iCloud Drive too, but most people won't realize that because they think iCloud Drive is a Dropbox type of service. The OS doesn't even give a handy message to users about how to do that and the Game Center system preference pane is sparse with no information.
So you have to enable iCloud, Game Center, AND iCloud Drive. Not a good solution. Steam doesn't force anyone to sign up to online storage cloud.
The bigger problem I'm seeing is that Apple Arcade games download to the default Applications folder. You can't put them in folders like on iOS. You can't store your games on an external drive. If you quickly run out of disk space you have no choice but to delete games or perform some hacking scripts (that Apple frowns on) to change default directories.