I made my gamecenter account back when it first came out, and I made an Apple ID specifically for gamecenter. It was really weird that it let me do that, looking back now. So, having something like that takes away features such as having a status or photo. That never really bothered me, but now, I can't view any games on the "games tab". I click on a game (to view achievements, friend scores, etc.), and the app crashes. I've tried hard resets, signing in and out of my account, everything that I could think of. Nothing worked. I tested it with a normal Apple ID (one you would use to buy things on the AppStore and sign into iCloud etc.) gamecenter and everything worked fine. The problem is, none of my achievements, friends, scores, name, etc. transfer over. I didn't think they would, but now I'm wondering if it's possible to transfer everything onto that Apple ID. I'm not very hopeful, but I thought I'd ask anyways. Also, apologies for the block of text, I'm to lazy to figure out where to use line breaks right now :/
I had this problem too, but factory reset fixed it. I know you aren't gonna want to do that, so make sure your on the latest iOS. And I understand about the line breaks.