I only use it because I play Smurfs Village with my niece and that's how she can visit my village and vice versa. (It's game center or facebook, so lesser of two evils)
It's rare that i'd get kicked on my touch - which is running iOS8, or was) but I'd have to sign in once or twice a week on my iPad , which is iOS7. Now I'm getting the annoying popups that freeze the game wanting me to log in via the popup, yet it doesn't work that way. I continually get 'cannot connect to server', but if I sign in via the settings, it seems to work.
If it is indeed on Apple's end, all folks can do is use work arounds or stop playing game center games.
I do agree with you that it was somehow triggered at the time of the iOS9 release. Because that's precisely when mine started messing up.