Show an ongoing operation when checking for updates

[peppy]

Addresses #30950.

Thought this would be a quick one, but during testing I noticed an unobserved exception could lead to a failure (even on master). Also, if the settings overlay was closed during the update check, the Schedule in the ContinueWith would not run (as the settings panel is not present).

I've fixed both issues by using the (relatively unused) SynchronizationContext presence, which means the continuation will be automatically scheduled back to the update thread (since it was triggered from that thread).

Can be tested using some iteration of:

diff --git a/osu.Game/Updater/UpdateManager.cs b/osu.Game/Updater/UpdateManager.cs
index c114e3a8d0..d9aa12fa46 100644
--- a/osu.Game/Updater/UpdateManager.cs
+++ b/osu.Game/Updater/UpdateManager.cs
@@ -26,9 +26,7 @@ public partial class UpdateManager : CompositeDrawable
         /// <summary>
         /// Whether this UpdateManager should be or is capable of checking for updates.
         /// </summary>
-        public bool CanCheckForUpdate => game.IsDeployedBuild &&
-                                         // only implementations will actually check for updates.
-                                         GetType() != typeof(UpdateManager);
+        public bool CanCheckForUpdate => true;
 
         [Resolved]
         private OsuConfigManager config { get; set; } = null!;
@@ -70,13 +68,15 @@ protected override void LoadComplete()
 
         public async Task<bool> CheckForUpdateAsync()
         {
+            await Task.Delay(3000).ConfigureAwait(false);
+
             if (!CanCheckForUpdate)
                 return false;
 
             Task<bool> waitTask;
 
             lock (updateTaskLock)
-                waitTask = (updateCheckTask ??= PerformUpdateCheck());
+                waitTask = updateCheckTask ??= PerformUpdateCheck();
 
             bool hasUpdates = await waitTask.ConfigureAwait(false);