OK guys ... so I did the iCloud backup for the important apps only [unique data that can't be retrieved from the Cloud], then restored it to my iPhone (Which took forever to download all the apps from the Appstore [specially whatsapp and it's backup]), and then tested the Game Center...
At the beginning: opening the Game Center app was still showing the blank page, and tapping <Game Center> in <Settings> was hanging, but after like 5 seconds, I was able to get into the Settings of Game Center; Into which I signed out from my previous user (which was my user only as in USER instead of USER@icloud.com), then I tried to sign with the same user, it didn't let me ... so I signed with USER@icloud.com, which worked ...
After few attempts, I was able to actually login to the Game Center app and get my previous data !!!
However, Game Center still takes time to open, around a minute or two, where before I remember leaving it for more than 30 mins without any progress
Check the images for the proof ...
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Good luck testing the same method 🙂