CmdPal: Use Shell API to determine the default browser in WebSearch (#43339)

## Summary of the Pull Request

This PR introduces a new method for determining the default browser
using the Windows Shell API. The new provider selects the browser
associated with the HTTPS protocol (falling back to HTTP if necessary).
The original implementation is retained as a fallback for now, and the
codebase is prepared for future extensions (e.g., manual default-browser
selection).

As a flyby, it also fixes an issue where commands continued showing the
previous browser name if the user changed their default browser while
the Command Palette was running.

## One-liner for change log

Fixed default browser selection in the Web Search built-in extension.

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## PR Checklist

- [x] Closes: #42343
- [ ] **Communication:** I've discussed this with core contributors
already. If the work hasn't been agreed, this work might be rejected
- [x] **Tests:** Added/updated and all pass
- [ ] **Localization:** All end-user-facing strings can be localized
- [ ] **Dev docs:** Added/updated
- [ ] **New binaries:** Added on the required places
- [ ] [JSON for
signing](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ESRPSigning_core.json)
for new binaries
- [ ] [WXS for
installer](https://github.com/microsoft/PowerToys/blob/main/installer/PowerToysSetup/Product.wxs)
for new binaries and localization folder
- [ ] [YML for CI
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ci/templates/build-powertoys-steps.yml)
for new test projects
- [ ] [YML for signed
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/release.yml)
- [ ] **Documentation updated:** If checked, please file a pull request
on [our docs
repo](https://github.com/MicrosoftDocs/windows-uwp/tree/docs/hub/powertoys)
and link it here: #xxx

<!-- Provide a more detailed description of the PR, other things fixed,
or any additional comments/features here -->
## Detailed Description of the Pull Request / Additional comments

<!-- Describe how you validated the behavior. Add automated tests
wherever possible, but list manual validation steps taken as well -->
## Validation Steps Performed

src/modules/cmdpal/Tests/Microsoft.CmdPal.Ext.WebSearch.UnitTests/MockBrowserInfoService.cs

@@ -0,0 +1,12 @@
+// Copyright (c) Microsoft Corporation
+// The Microsoft Corporation licenses this file to you under the MIT license.
+// See the LICENSE file in the project root for more information.
+
+using Microsoft.CmdPal.Ext.WebSearch.Helpers.Browser;
+
+namespace Microsoft.CmdPal.Ext.WebSearch.UnitTests;
+
+public class MockBrowserInfoService : IBrowserInfoService
+{
+    public BrowserInfo GetDefaultBrowser() => new() { Name = "mocked browser", Path = "C:\\mockery\\mock.exe" };
+}

src/modules/cmdpal/Tests/Microsoft.CmdPal.Ext.WebSearch.UnitTests/QueryTests.cs

@@ -26,8 +26,9 @@ public class QueryTests : CommandPaletteUnitTestBase
     {
         // Setup
         var settings = new MockSettingsInterface();
+        var browserInfoService = new MockBrowserInfoService();
 
-        var page = new WebSearchListPage(settings);
+        var page = new WebSearchListPage(settings, browserInfoService);
 
         // Act
         page.UpdateSearchText(string.Empty, query);
@@ -55,8 +56,9 @@ public class QueryTests : CommandPaletteUnitTestBase
         };
 
         var settings = new MockSettingsInterface(mockHistory: mockHistoryItems, historyItemCount: 5);
+        var browserInfoService = new MockBrowserInfoService();
 
-        var page = new WebSearchListPage(settings);
+        var page = new WebSearchListPage(settings, browserInfoService);
 
         // Act
         page.UpdateSearchText("abcdef", string.Empty);
@@ -90,8 +92,9 @@ public class QueryTests : CommandPaletteUnitTestBase
         };
 
         var settings = new MockSettingsInterface(mockHistory: mockHistoryItems, historyItemCount: 5);
+        var browserInfoService = new MockBrowserInfoService();
 
-        var page = new WebSearchListPage(settings);
+        var page = new WebSearchListPage(settings, browserInfoService);
 
         mockHistoryItems.Add(new HistoryItem("another search5", DateTime.Parse("2024-01-06 13:00:00", CultureInfo.CurrentCulture)));
 
@@ -123,8 +126,9 @@ public class QueryTests : CommandPaletteUnitTestBase
         };
 
         var settings = new MockSettingsInterface(mockHistory: mockHistoryItems, historyItemCount: 0);
+        var browserInfoService = new MockBrowserInfoService();
 
-        var page = new WebSearchListPage(settings);
+        var page = new WebSearchListPage(settings, browserInfoService);
 
         // Act
         page.UpdateSearchText("abcdef", string.Empty);

src/modules/cmdpal/Tests/Microsoft.CmdPal.Ext.WebSearch.UnitTests/SettingsManagerTests.cs

@@ -20,7 +20,9 @@ public class SettingsManagerTests : CommandPaletteUnitTestBase
     {
         // Setup
         var settings = new MockSettingsInterface(historyItemCount: 5);
-        var page = new WebSearchListPage(settings);
+        var browserInfoService = new MockBrowserInfoService();
+
+        var page = new WebSearchListPage(settings, browserInfoService);
 
         var eventRaised = false;