I have Game Center working with Ejecta on several games. It seems the Game Center function does not work at all with the simulator, you need to deploy to an actual device.
The device need to be registered as a development device with the correct provisioning profiles installed.
Make sure you are logged in to Game Center on the device as well, as this seems to help. Once the game starts, a soft authenticate will usually work, and the GC welcome notification should momentarily pop down from the top of the screen.
One thing to check is that the app has the GameCenter capability added to the info.plist. This is done automatically by Xcode if you click the GameCenter slider in the 'Capability' tab.
Of course you need to set up Game Center leaderboards in iTunes connect first, I guess you've already done that.