Was having trouble with Gamecenter too but I literally just got it working myself! Looks like you're going through the same process as I am, with exporting from Construct as Cordova and then using XDK after that. It wasn't working for me and I read that I should be seeing a Sandbox setting in the Gamecenter settings. But that wasn't there. I then read about the Testflight app and wasn't sure if it was the solution, but I installed it anyway and after that the Sandbox setting appeared, which I then switched on.
So I then loaded up my game and Gamecenter asked for authentication! I logged in, which then made the leaderboard show up once I clicked my leaderboard button.
So try what I did if you haven't already. And it may seem like a stupid question but you are actually testing on an iOS device right?
Edit: I may also add that I'm using the Adhoc profile and installing the game with iTunes, don't know if that would make a difference though.