[Docs Labs] Experimenting with sidebar search

[Oxyjun]

Summary

Screenshots (optional)

Documentation checklist

• Is there a changelog entry (guidelines)? If you don't add one for something awesome and new (however small) — how will our customers find out? Changelogs are automatically posted to RSS feeds, the Discord, and X.
• The documentation style guide has been adhered to.
• If a larger change - such as adding a new page- an issue has been opened in relation to any incorrect or out of date information that this PR fixes.
• Files which have changed name or location have been allocated redirects.

[github-actions]

This pull request requires reviews from CODEOWNERS as it changes files that match the following patterns:

Pattern	Owners
*.astro	@cloudflare/developer-advocacy, @kristianfreeman, @kodster28, @pedrosousa, @haleycode, @GregBrimble, @KianNH, @WalshyDev

[github-actions]

Preview URL: https://00ae49be.preview.developers.cloudflare.com
Preview Branch URL: https://jun-docslab-sidebar-search.preview.developers.cloudflare.com

[Oxyjun]

This had unforeseen limitations, taking over here #24686 with better logic.