As pointed out by MacRumors' sister site TouchArcade, Game Center is still not completely gone away as a service. In the release notes for the first beta of iOS 10, Apple explains that if a developer wants Game Center-like leaderboards in their app, they will have to implement such features themselves.
• The Game Center app has been removed. If your game implements GameKit features, it must also implement the interface behavior necessary for the user to see these features. For example, if your game supports leaderboards, it could present a GKGameCenterViewController object or read the data directly from Game Center to implement a custom user interface.When it was introduced, Game Center was presented as a way for users to visit a centralized hub of all of their gaming applications on iOS, although not every game supported Apple's app. Game Center provided profiles, achievements, leaderboards, status updates on friends' most recently played games, and a specialized "Turns" tab to keep track of games that used asynchronous multiplayer features.
• A new account type, implemented by the GKCloudPlayer class, supports iCloud-only game accounts.
• Game Center provides a new generalized solution for managing persistent storage of data on Game Center. A game session (GKGameSession) has a list of players who are the session’s participants. Your game’s implementation defines when and how a participant stores or retrieves data from the server or exchanges data between players. Game sessions can often replace existing turn-based matches, real-time matches, and persistent save games, and also enable other models of interaction between participants.
Apple appears to be addressing multiple user issues with its stock apps in the iOS 10 developer beta, allowing for the removal of over 20 of its first party applications, a long-requested feature many users have been asking for over the past few years.
Read up on more iOS 10 features in our tidbits post here.
Update: As one user has pointed out on Twitter, "Game Center" was mentioned in name during the watchOS 3 section of the keynote. What this means for the future of the service, on either iOS or watchOS, is still unclear.
Update 2: Apple has confirmed to TechCrunch that "Game Center will continue as a service, but it will no longer be available as a standalone application."