While the app can still be found in the Settings of iOS 10, the Game Center icon no longer shows up on the Home screen. This does not mean that developers can no longer use Game Center for apps, but that they will need to implement a custom user interface with GameKit.
Personally, I feel like Game Center never really made a big splash upon iOS users. The SDK is great for unifying any online connections in games. However, it seems achievements made way onto iOS during the same time ones on other platforms began to lose their touch.
In addition, Apple's GameKit now supports up to four MFi controllers at the same time and ReplayKit, which allows for live streaming of games on iOS.
Source: Apple
