I have tried the above thanks.
So I go to settings > Game Center > sign out.
And it all signs out successfully, and stays that way for a few hours/days.
After signing out, whenever I launch games, it pops up the attached welcome to Game Center screen. I tap on cancel and it goes away. This keeps happening until it stops happening.. but when it stops, if I go back to settings > Game Center, I find I’m logged in again. Despite cancelling all log in prompts and not proactively logging in at all.
It happens on my iPhone and iPad. All fully up-to-date, the iPad has recently been wiped clean to try resolve another issue - and it still does it.
I want to log out, stay logged out and not be promoted to log in when I launch games! 