我有一个固定。
我在我的SwiftUI应用程序Sound Matcher中成功地使用了Game Center。代码片段如下。
代码并没有完全遵循SwiftUI的声明性理念，但它运行得很好。我向SceneDelegate和ContentView添加了代码片段，并使用了一个GameKitHelper类，类似于托马斯为他的测试应用创建的类。我的版本基于我在raywenderlich.com上找到的代码。
实际上我第一次尝试使用符合UIViewControllerRepresentable的结构体，遵循与bg2b相同的思路，但它一直抱怨游戏中心视图控制器需要模态呈现。最终我放弃了，并尝试了目前更成功的方法。
适用于SwiftUI 1.0和iOS 13

class SceneDelegate: UIResponder, UIWindowSceneDelegate {

    let contentView = ContentView()
        .environmentObject(GameKitHelper.sharedInstance) // publish enabled state

func scene(_ scene: UIScene, willConnectTo session: UISceneSession, 
            options connectionOptions: UIScene.ConnectionOptions) {
    
    if let windowScene = scene as? UIWindowScene {
        let window = UIWindow(windowScene: windowScene)

        window.rootViewController = UIHostingController(rootView: contentView)
        // new code to create listeners for the messages
        // you will be sending later
        PopupControllerMessage.PresentAuthentication
             .addHandlerForNotification(
                 self, 
                 handler: #selector(SceneDelegate
                     .showAuthenticationViewController))
                
        PopupControllerMessage.GameCenter
            .addHandlerForNotification(
                self, 
                handler: #selector(SceneDelegate
                   .showGameCenterViewController))

        // now we are back to the standard template
        // generated when your project was created
        self.window = window
        window.makeKeyAndVisible()
    }
}
// pop's up the leaderboard and achievement screen
@objc func showGameCenterViewController() {
         if let gameCenterViewController =
             GameKitHelper.sharedInstance.gameCenterViewController {
                    self.window?.rootViewController?.present(
                         gameCenterViewController,
                         animated: true,
                         completion: nil)
         }
  
}
// pop's up the authentication screen
@objc func showAuthenticationViewController() {
    if let authenticationViewController =
        GameKitHelper.sharedInstance.authenticationViewController {
              
           self.window?.rootViewController?.present(
                authenticationViewController, animated: true)
                { GameKitHelper.sharedInstance.enabled  = 
                  GameKitHelper.sharedInstance.gameCenterEnabled }
    }
  }
}

// content you want your app to display goes here
struct ContentView: View {
  

@EnvironmentObject var gameCenter : GameKitHelper
@State private var isShowingGameCenter = false { didSet { 
                        PopupControllerMessage
                           .GameCenter
                           .postNotification() }}
     
var body: some View { 
    VStack {
        if self.gameCenter.enabled
             { 
            Button(action:{ self.isShowingGameCenter.toggle()})
                { Text(
                  "Press to show leaderboards and achievements")}
             } 
        // The authentication popup will appear when you first enter
        // the view            
        }.onAppear() {GameKitHelper.sharedInstance
                               .authenticateLocalPlayer()}
    }
}

import GameKit
import UIKit
 
// Messages sent using the Notification Center to trigger 
// Game Center's Popup screen
public enum PopupControllerMessage : String
{
 case PresentAuthentication = "PresentAuthenticationViewController"
 case GameCenter = "GameCenterViewController"
}

extension PopupControllerMessage
{
  public func postNotification() {
     NotificationCenter.default.post(
        name: Notification.Name(rawValue: self.rawValue),
        object: self)
  }
    
  public func addHandlerForNotification(_ observer: Any, 
                                        handler: Selector) {
     NotificationCenter.default .
          addObserver(observer, selector: handler, name:
            NSNotification.Name(rawValue: self.rawValue), object: nil)
  }
    
}

// based on code from raywenderlich.com
// helper class to make interacting with the Game Center easier

open class GameKitHelper: NSObject,  ObservableObject,  GKGameCenterControllerDelegate  {
    public var authenticationViewController: UIViewController?
    public var lastError: Error?

private static let _singleton = GameKitHelper()
public class var sharedInstance: GameKitHelper {
    return GameKitHelper._singleton
}

private override init() {
    super.init()
}
@Published public var enabled :Bool = false
   
public var  gameCenterEnabled : Bool { 
                     return GKLocalPlayer.local.isAuthenticated }

    public func authenticateLocalPlayer () {
        let localPlayer = GKLocalPlayer.local
        localPlayer.authenticateHandler = {(viewController, error) in
           
            self.lastError = error as NSError?
             self.enabled = GKLocalPlayer.local.isAuthenticated
            if viewController != nil {
                self.authenticationViewController = viewController                  
                PopupControllerMessage
                   .PresentAuthentication
                   .postNotification()
            }
        }
    }
   
    public var gameCenterViewController : GKGameCenterViewController? { get {
         
         guard gameCenterEnabled else {  
                  print("Local player is not authenticated")
                  return nil }
        
         let gameCenterViewController = GKGameCenterViewController()
         
         gameCenterViewController.gameCenterDelegate = self
         
         gameCenterViewController.viewState = .achievements
         
         return gameCenterViewController
        }}
    
    open func gameCenterViewControllerDidFinish(_ 
                gameCenterViewController: GKGameCenterViewController) {
   
        gameCenterViewController.dismiss(
                      animated: true, completion: nil)
    }
   
}

更新：对于SwiftUI 2.0和iOS 14，代码更容易

import GameKit

enum Authenticate
{
   static func user() {
       let localPlayer = GKLocalPlayer.local
       localPlayer.authenticateHandler = { _, error in
           guard error == nil else {
               print(error?.localizedDescription ?? "")
               return
           }
           GKAccessPoint.shared.location = .topLeading
           GKAccessPoint.shared.isActive =      
                            localPlayer.isAuthenticated
       }
   }
}

import SwiftUI

// content you want your app to display goes here
struct ContentView: View { 
     
var body: some View { 
      Text( "Start Game") 
        // The authentication popup will appear when you first enter
        // the view            
        }.onAppear() { Authenticate.user()}
    }
}

在此为您提供部分答案。如果设备（或模拟器）已登录iCloud并且已在设置中启用GameCenter，我可以下载排行榜分数并在SwiftUI列表中显示它们。如果不是这样，我没有尝试显示gameCenter身份验证视图控制器。
感谢您的问题中的代码。我使用了您的GameCenterManager（），但将其放在了我的AppDelegate中：

let gameCenter = GameCenterManager()

下面是我的ShowRankings。swift SwiftUI视图。我能够成功地验证并获得分数。但我仍然有“异常”。我第一次运行这个（在模拟器中）我收到预期的“User is not logged into Game Center”（用户未登录到游戏中心）错误，指示您的GameCenterManager中的ViewController不是零（我甚至从来没有尝试过显示它）。但是我能够成功地获得分数并将它们显示在列表中。

import SwiftUI
import GameKit

struct ShowRankings: View {
    let appDelegate = UIApplication.shared.delegate as! AppDelegate

    let leaderBoard = GKLeaderboard()
    @State var scores: [GKScore] = []
    var body: some View {
        VStack {
            Button(action: {
                self.updateLeader()
            }) {
                Text("Refresh leaderboard")
            }
            List(scores, id: \.self) { score in
                Text("\(score.player.alias) \(score.value)")
            }
        }.onAppear() {
            self.appDelegate.gameCenter.authenticateLocalPlayer()
            self.updateLeader()
        }
    }
    func updateLeader() {
        let leaderBoard: GKLeaderboard = GKLeaderboard()
        leaderBoard.identifier = "YOUR_LEADERBOARD_ID_HERE"
        leaderBoard.timeScope = .allTime
        leaderBoard.loadScores { (scores, error) in
            if let error = error {
                debugPrint("leaderboard loadScores error \(error)")
            } else {
                guard let scores = scores else { return }
                self.scores = scores
            }
        }
    }
}

另一种解决方案是为GameCenter创建一个UIViewControllerRepresentable，它需要一个排行榜ID来打开。这使得打开特定排行榜变得简单。

public struct GameCenterView: UIViewControllerRepresentable {
    let viewController: GKGameCenterViewController
    
    public init(leaderboardID : String?) {
        
        if leaderboardID != nil {
            self.viewController = GKGameCenterViewController(leaderboardID: leaderboardID!, playerScope: GKLeaderboard.PlayerScope.global, timeScope: GKLeaderboard.TimeScope.allTime)
        }
        else{
            self.viewController = GKGameCenterViewController(state: GKGameCenterViewControllerState.leaderboards)
        }
        
    }
    
    public func makeUIViewController(context: Context) -> GKGameCenterViewController {
        let gkVC = viewController
        gkVC.gameCenterDelegate = context.coordinator
        return gkVC
    }
    
    public func updateUIViewController(_ uiViewController: GKGameCenterViewController, context: Context) {
        return
    }
    
    public func makeCoordinator() -> GKCoordinator {
        return GKCoordinator(self)
    }
}

public class GKCoordinator: NSObject, GKGameCenterControllerDelegate {
    var view: GameCenterView
    
    init(_ gkView: GameCenterView) {
        self.view = gkView
    }
    
    public func gameCenterViewControllerDidFinish(_ gameCenterViewController: GKGameCenterViewController) {
        gameCenterViewController.dismiss(animated: true, completion: nil)
    }
}

要使用，只需在需要显示排行榜的位置添加以下。

GameCenterView(leaderboardID: "leaderBoardID")