Toggling off GameCenter in iCloud doesn't do much for you, it just prevents your game scores and statuses from being uniform across your devices.
You have to sign out of the GameCenter, per instructions below. Before you tap on the Sign Out button you may want to look just above at Connect with Friends where a list of games on your device might appear (I don't have any game so I cannot actually verify that).
This is from https://www.makeuseof.com/tag/use-disable-game-center-mac-ios/
How to Sign Out of Game Center
Game Center is built into the operating system on your iPhone or Mac, so you can't delete or remove it. But if you don't want to use Game Center on your device anymore, you can sign out and turn the feature off.
This doesn't delete your Game Center account, so you can sign in again at any time to recover your saved games. Unfortunately, there's no way to delete your Game Center account without deleting your Apple ID.
Sign Out on an iPhone, iPad, or iPod touch
Open the Game Center settings again, then scroll down and tap the Sign Out button. This automatically signs you out of Game Center and disables the feature.