Remove unused properties in AP (#43564)

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## Summary of the Pull Request
This pull request removes legacy provider configuration migration logic
and associated data structures from the Advanced Paste AI provider
settings. The changes simplify the codebase by eliminating support for
legacy provider configuration snapshots and related migration methods,
focusing configuration management on the current provider model.

## PR Checklist

- [ ] Closes: #xxx
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## Detailed Description of the Pull Request / Additional comments

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## Validation Steps Performed

---------

Signed-off-by: Shawn Yuan (from Dev Box) <shuaiyuan@microsoft.com>

src/settings-ui/Settings.UI/ViewModels/AdvancedPasteViewModel.cs

@@ -191,6 +191,18 @@ namespace Microsoft.PowerToys.Settings.UI.ViewModels
                 return;
             }
 
+            PasswordCredential legacyCredential = TryGetLegacyOpenAICredential();
+
+            if (legacyCredential is null)
+            {
+                if (legacyAdvancedAIConsumed)
+                {
+                    SaveAndNotifySettings();
+                }
+
+                return;
+            }
+
             var configuration = properties.PasteAIConfiguration;
             if (configuration is null)
             {
@@ -198,28 +210,11 @@ namespace Microsoft.PowerToys.Settings.UI.ViewModels
                 properties.PasteAIConfiguration = configuration;
             }
 
-            bool hasLegacyProviders = configuration.LegacyProviderConfigurations is { Count: > 0 };
-            PasswordCredential legacyCredential = TryGetLegacyOpenAICredential();
-
-            if (!hasLegacyProviders && legacyCredential is null && !legacyAdvancedAIConsumed)
-            {
-                return;
-            }
-
             bool configurationUpdated = false;
 
-            if (hasLegacyProviders)
-            {
-                configurationUpdated |= AdvancedPasteMigrationHelper.MigrateLegacyProviderConfigurations(configuration);
-            }
-
-            PasteAIProviderDefinition openAIProvider = null;
-            if (legacyCredential is not null || hasLegacyProviders || legacyAdvancedAIConsumed)
-            {
-                var ensureResult = AdvancedPasteMigrationHelper.EnsureOpenAIProvider(configuration);
-                openAIProvider = ensureResult.Provider;
-                configurationUpdated |= ensureResult.Updated;
-            }
+            var ensureResult = AdvancedPasteMigrationHelper.EnsureOpenAIProvider(configuration);
+            PasteAIProviderDefinition openAIProvider = ensureResult.Provider;
+            configurationUpdated |= ensureResult.Updated;
 
             if (legacyAdvancedAIConsumed && openAIProvider is not null && openAIProvider.EnableAdvancedAI != legacyAdvancedAIEnabled)
             {
@@ -233,7 +228,7 @@ namespace Microsoft.PowerToys.Settings.UI.ViewModels
                 RemoveLegacyOpenAICredential();
             }
 
-            bool shouldEnableAI = legacyCredential is not null;
+            const bool shouldEnableAI = true;
             bool enabledChanged = false;
             if (properties.IsAIEnabled != shouldEnableAI)
             {
@@ -1248,11 +1243,6 @@ namespace Microsoft.PowerToys.Settings.UI.ViewModels
                 return true;
             }
 
-            if (current.UseSharedCredentials != incoming.UseSharedCredentials)
-            {
-                return true;
-            }
-
             var currentProviders = current.Providers ?? new ObservableCollection<PasteAIProviderDefinition>();
             var incomingProviders = incoming.Providers ?? new ObservableCollection<PasteAIProviderDefinition>();
 
@@ -1408,8 +1398,7 @@ namespace Microsoft.PowerToys.Settings.UI.ViewModels
                 return;
             }
 
-            if (string.Equals(e.PropertyName, nameof(PasteAIConfiguration.ActiveProviderId), StringComparison.Ordinal)
-                || string.Equals(e.PropertyName, nameof(PasteAIConfiguration.UseSharedCredentials), StringComparison.Ordinal))
+            if (string.Equals(e.PropertyName, nameof(PasteAIConfiguration.ActiveProviderId), StringComparison.Ordinal))
             {
                 SaveAndNotifySettings();
             }
@@ -1426,15 +1415,7 @@ namespace Microsoft.PowerToys.Settings.UI.ViewModels
 
             pasteConfig.Providers ??= new ObservableCollection<PasteAIProviderDefinition>();
 
-            bool configurationUpdated = AdvancedPasteMigrationHelper.MigrateLegacyProviderConfigurations(pasteConfig);
-
             SubscribeToPasteAIProviders(pasteConfig);
-
-            if (configurationUpdated)
-            {
-                SaveAndNotifySettings();
-                OnPropertyChanged(nameof(PasteAIConfiguration));
-            }
         }
 
         private static string RetrieveCredentialValue(string credentialResource, string credentialUserName)