Well....it's fixed!
All you need to do is strip naked, go outside, hold your affected device above your head and spin around six times chanting "Apple is god".
Then throw your device on a pile of five day old grass clippings, set fire to the pile, face due north, hop on one leg holding your left arm out at a 90 degree angle and your right arm resting on your hip and this time chant "I'm a little tea pot short and stout. Here is my handle. Here is my spout."
By this time two men in white uniforms should arrive and take you away and you will no longer get affected by the Game Center bug.
In all seriousness this bug is getting worse. Every day Game Center is not working for me and every day I have to do the Airplane Method, sometimes two or three times to get it to work, and then it only remains working for a short time.
Apple seems to have no interest in fixing it as people have been posting in all sorts of forums about the problem for years.
I have decided that I will simply no longer purchase any games, or games with iap's that require Game Center to work.
It seems to be the only sure way to be rid of the problem.
Bad luck for developers though.
Absolutely pathetic service from Apple.