Implement tree browser search panel for file tree browser

Context

This is a follow-up to #543753 (closed), which addressed the issue of client-side filtering in the file tree browser. In !213515 (merged), we implemented Iteration 1: triggering the global search modal when users click the search button or press the f keyboard shortcut.

This issue tracks Iteration 2: implementing a dedicated tree browser search panel that displays search results closer to the search field itself, providing better context and user experience.

Problem

While the global search modal (Iteration 1) provides full repository search functionality, the search results appear in a separate modal that takes users away from the file tree context. Users would benefit from having search results displayed inline, closer to the search field, maintaining their focus within the file tree browser.

Design Reference

See the design proposal in #543753 (closed) (note 2804696436):

Implementation Considerations

• Reuse existing file search backend functionality
• Create a new search results component that can be positioned near the search input
• Ensure keyboard navigation works seamlessly
• Consider how this interacts with the existing f keyboard shortcut
• Maintain accessibility standards

Related

• Parent issue: #543753 (closed)
• Iteration 1 implementation: !213515 (merged)