Under iOS 12, I had signed out of Game Center and it didn’t bother me or pop up.
Under iOS 13 / iPadOS, even when you sign out of Game Center (and it shows as dimmed out in the Blitz settings area), it will still pop up, asking you to sign in, when you start the game.
To get around this, you need also need to go to the Notifications area of your device, find "Games" (not Game Center), and turn the Allow Notifications switch off as well.
This should prevent Game Center from popping up and asking you to login each time Blitz starts.