So I heard that recently Apple has decided to remove the standalone app for game center and replace it with all this gimmicky stuff. I was wondering how am I supposed to transfer my clash royale account to a new device without game center?
You can still login into Game Center inside the Settings. On a new device you login into Game Center with the same Apple ID as your old device and your done.
So all the synchronization of high scores, progress and achievements still works. Apple decided to focus on the background technology for Game developers instead of the stand alone app.