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

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or any additional comments/features here -->
## Detailed Description of the Pull Request / Additional comments

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## Validation Steps Performed

.github/actions/spell-check/expect.txt

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