Fix homepage search after query navigation

[mathix420]

Summary

• synchronize the homepage search and sort controls when client-side navigation changes URL query parameters
• initialize the quick-search async-data entry even when the initial query is empty, so later query changes execute the request
• preserve the existing indexed PostgreSQL full-text search behavior

Root cause

Nuxt reuses the homepage component when only query parameters change, but the local search state was initialized only once. After synchronizing that state, a second Nuxt async-data edge case remained: immediate: false created a new fetch key for the updated query without executing it, while the query watcher was suppressed during that key transition.

Impact

Homepage searches initiated through typing, WebMCP, or browser back/forward navigation now fetch and render results without requiring a page refresh.

Validation

• bunx vize fmt --check app/pages/index.vue
• bunx vize lint app/pages/index.vue (passes with six pre-existing template-style warnings)
• Chrome regression: client-side navigation from / to /?search=cheaper requested /api/search/quick?q=cheaper and rendered Cheaper access to clean water
• local API verification returned the expected issue and solution matches for cheaper

[github-actions]

🧹 Code quality

Check		Summary
vize lint	✅	0 errors · 314 warnings
vize format	✅	all files formatted
fallow audit	✅	pass · 0 dead-code · 0 complexity · 0 dupes (4 files)

_{updated for b82c44a · run #299}