Fix process exit after automatic browser launch

[mortenn]

Summary

• Avoid showing the picker window after an automatic browser selection already launched the target browser.
• Route automatic-launch shutdown through the normal app shutdown path so background tasks are cancelled consistently.
• Fixes Process does not exit after sending url to browser #227.

Test plan

• dotnet build "src/BrowserPicker.App/BrowserPicker.App.csproj" -p:Version=1.0.0
• Manually verify the Outlook/Slack link path no longer leaves BrowserPicker running after auto-launch.