Inside iOS 10: With Game Center app gone, invites are managed by Messages
Beginning with iOS 10, Apple's Game Center service will no longer have its own dedicated application. Instead, actions like adding friends or inviting them to a game will be handled by individual games themselves, routed through Apple's expanded Messages app.

As detailed by Apple at a session during its Worldwide Developers Conference last week, games will have access to new tools for inviting friends to multiplayer games via the Messages app.
Apple told developers that those already using existing Game Center invitation APIs won't need to make any code changes for iOS 10. Messages-based invites will automatically replace the old method of inviting friends to play, and anyone can be invited via the Messages app and iCloud.

In one demonstration, Apple software engineer Megan Gardner showed a player sending an invite to play the popular title "Doodle Jump." The invitation was sent as a "rich link" attachment in an iMessage to another iPhone user, with the option to add additional text to the message.
Invitations can be sent to players saved in a user's contacts, recent players who were seen online, or nearby players on a local network playing the same title. Players with legacy Game Center accounts will also receive notifications from Messages-based invites.

For users on iOS 10, once a Game Center invitation has been received in Messages, tapping on a rich link will take the user directly into the game. If they do not have that particular title installed, the link will instead open the game's listing on the iOS App Store.
In addition to new ways of inviting players, Game Center has also added support for Apple Watch and watchOS 3.
Game Center management within apps has allowed Apple to get rid of the dedicated Game Center app starting with iOS 10. The multiplayer gaming network has had its own standalone app on iPhone and iPad since 2010.
For more, see AppleInsider's ongoing Inside iOS 10 series. iOS 10 is currently available to developers for beta testing, and will be finalized for a free public launch this fall.

As detailed by Apple at a session during its Worldwide Developers Conference last week, games will have access to new tools for inviting friends to multiplayer games via the Messages app.
Apple told developers that those already using existing Game Center invitation APIs won't need to make any code changes for iOS 10. Messages-based invites will automatically replace the old method of inviting friends to play, and anyone can be invited via the Messages app and iCloud.

In one demonstration, Apple software engineer Megan Gardner showed a player sending an invite to play the popular title "Doodle Jump." The invitation was sent as a "rich link" attachment in an iMessage to another iPhone user, with the option to add additional text to the message.
Invitations can be sent to players saved in a user's contacts, recent players who were seen online, or nearby players on a local network playing the same title. Players with legacy Game Center accounts will also receive notifications from Messages-based invites.

For users on iOS 10, once a Game Center invitation has been received in Messages, tapping on a rich link will take the user directly into the game. If they do not have that particular title installed, the link will instead open the game's listing on the iOS App Store.
In addition to new ways of inviting players, Game Center has also added support for Apple Watch and watchOS 3.
Game Center management within apps has allowed Apple to get rid of the dedicated Game Center app starting with iOS 10. The multiplayer gaming network has had its own standalone app on iPhone and iPad since 2010.
For more, see AppleInsider's ongoing Inside iOS 10 series. iOS 10 is currently available to developers for beta testing, and will be finalized for a free public launch this fall.
Comments
And I would not want to have to block people just to be never bothered by such garbage.
It looks like there's a template for people who don't want to make a custom design:
It won't be as easy to see an overview across all games but people might not have been doing that anyway. Apple will have access logs for achievements in-app and via the Game Center app and they likely discovered that few people use the app to interact with Game Center.
Using Messages for invitations is a logical way for people to invite their friends to play but people won't always want to be messaging people they want to play a game with, it could just be a random person on the other side of the world that they don't want to turn into a pen-pal. The outgoing message would be better as a semi-anonymous Game Center notification and not a message. It would make it easier to decline as a notification without feeling bad and without having to make up excuses for not wanting to play with message replies.
I play SimCity BuildIt regularly. The only ways to keep my progress on the Cloud is either through a FaceBook login, or via Game Center. I don't have a FaceBook login, so I use Game Center.
I hope the option to save on the Cloud is still provided by EA Sports.
As long as I don't have to figure out a way to avoid GC iMessages, RIP.