I'm not sure where everyone is at on this, but I spoke to Apple yesterday and I have a case open. They escalated my call three levels up and even then, it did not seem to be a problem that they knew about. The guy I'm dealing with has pushed it up further and called me back to day saying it is an issue they are aware of. What they are doing about it is still unclear.
The behavior I've been experiencing is that I log out of game center and eventually, when I enter a game, I'm logged back in, even though I did not login.
Between my call and their call back, I figured out how I am being logged back in. It would appear that whenever I make a purchase where I enter my password, if I check right before to make sure I'm logged out, then for sure I'm logged back in after. So any app purchasing logs me into game center.
Secondly, if I go to to "Settings" and "ITunes & App Stores" and click on my Apple ID and select "View Apple ID", I'm asked to enter my password and this also logs me into game center. I can't think of other instances where I use my Apple ID off the top of my head, but I would presume anytime I authenticate to anything on my phone using my Apple ID, it's used as an opportunity to throw me back, fully logged in to Game Center.
Sneaky I'd say. A programming error? Seems unlikely. I presume that upgrading the iOS probably has the same result. I have not checked since the last iOS upgrade whether any of the things I had turned off in GC might have been turned back on, but don't think so and hope not.
That's where I'm at with this. I encourage you to call their support. Actually I went onto the chat online. The first guy was useless (literally), but eventually realized it himself and got his supervisor. That wasn't much better, but at least he decided that someone would call me from level 3. This 3rd person was more sympathetic, but did not seem to have any idea that there was such an issue. But he managed to get someone higher up to take the call. There I got no acknowledgement that a problem was known, but I got lots of sympathy and a promise of follow-up. And so far, so good, the guy called me back in less than 24 hrs and I'm expecting more answers soon I hope.