I went and did that. I had previously the 'open 3 times, disable' routine and had not had the pop up in several days. After changing the settings you said and nothing else, the next time I opened my game I got the little box that said 'welcome back to the game center' which I thought was strange because I made absolutely sure to logout of Game Center and kill the app the last time I had the pop up...
It's the 2nd time that after doing settings changes related to the Game Center, I find myself with that welcome screen after opening a game. First time I thought I must have done something wrong or missed something. BUT THIS TIME I know for a fact I had logged out of Game Center and had never gone back in. Had only changed the settings as you suggest.
So does that mean that this is set up in such a way that you automatically get relogged in given certain conditions. That seems kind of crazy borderline fraudulent and totally a security hole if not simply a door wide open and with a big flashing neon sign that says: "Hackers Enter HERE"
I'm amazed that Apple did this. It reminds me of certain things that ran in the background in windows that sucked up all your resources. Or Adobe with incessant update popups in windows and osx that you can never get rid of short of removing the adobe software. But at least you can remove it if you don't like it, unlike Game Center.