Replace Astro built-in search with Mixedbread AI search

[schickling]

Summary

Replace the current Astro built-in search functionality with Mixedbread AI's semantic search to provide better, more intelligent search results for LiveStore documentation.

Context

Maxwell is currently working on a similar implementation for the Effect docs, making this a perfect opportunity to leverage that work and bring enhanced search capabilities to LiveStore.

Proposed Implementation

• Replace Astro's default search with Mixedbread AI integration
• Index LiveStore documentation content for semantic search
• Implement search UI that provides more relevant and contextual results
• Ensure search works across all documentation sections

Future Enhancements

Once the documentation search is implemented, we can explore:

• Indexing LiveStore Discord conversations via the LiveStore Discord bot
• Cross-platform search across docs and community discussions

Benefits

• Better relevance: Semantic search understands context and intent
• Improved UX: More accurate results for technical queries
• Knowledge reuse: Leverage Maxwell's work on Effect docs implementation
• Extensible: Foundation for future Discord integration