How Does / Can GameCenter Save to iCloud?

I'm trying to save game data across devices. I'm trying to understand if GameCenter has the ability to do this.. I was reading from the docs..

https://developer.apple.com/library/archive/documentation/NetworkingInternet/Conceptual/GameKit_Guide/SavedGames/SavedGames.html

To provide saved game data between devices, developers need to create their own solutions, either using their own servers or using a secondary service, such as Facebook. Using the new saved game methods found in GKSavedGame, GKLocalPlayer, and GKSavedGameListener classes, developers can easily save game data to iCloud.

However, when I fetch saved games on separate devices, I get different results. Both devices are using the same GameCenter and iCloud accounts. Note: When I delete the app locally on a device, I am able to retrieve the local GameCenter data.

   var localPlayer: GKLocalPlayer {
       return GKLocalPlayer.local
   }

  func authenticateLocalPlayer(completion: @escaping((_ gameCenterLoginVC: UIViewController?,_ gameCenterEnabled: Bool ) -> Void)) {
        localPlayer.authenticateHandler = { [weak self] (viewController, error) -> Void in
            guard let self = self else { return }
            if viewController != nil {
                completion(viewController, true)
            } else if self.localPlayer.isAuthenticated {
                self.gcEnabled = true
                self.fetchSavedGames(completion: { (success) in
                    completion(nil, true)
                })
            } else {
                completion(nil, false)
            }
        }
    }

    private func fetchSavedGames(completion: @escaping((_ success: Bool) -> Void)) {
        localPlayer.fetchSavedGames { (savedGames, error) in
            // savedGames here has different results across devices.
            guard let savedGame = savedGames?.sorted(by: { (game1, game2) -> Bool in
                game1.modificationDate ?? Date.init(timeIntervalSince1970: 0) > game2.modificationDate ?? Date.init(timeIntervalSince1970: 0)
            }).first else {
                completion(false)
                return
            }
            savedGame.loadData(completionHandler: { (data, error) in
                guard let data = data, error == nil else {
                    completion(false)
                    return
                }
                User.shared.loadUserWith(data: data)
                completion(true)
            })
        }
    }