Why did my Game Center app disappear?

Apple discontinued the standalone Game Center app with the release of iOS 10. Third-party games can still utilize Game Center features from GameKit, such as leaderboards, but they will need to be accessed within the app instead of the standalone Game Center app. In short, Game Center features still exist in iOS games, but the Game Center app is no longer available.

You can read more about the Game Center changes in Apple's iOS 10 developer release notes under the GameKit section.