docs: S2 docs high priority fixes from testing (#9220)
* fix skeleton title for index page
* remove Home link from Overview section
* convert relative links in markdown from .html links to .md
* fix card logo centering in Safari
* add homepage links to mobile header
* fix blog video props to match originals
* improve search sections
* refine "All" tag display logic
* fix missing 'client side routing guides' and 'framework setup guides' links
* don't clear search field before navigation
* remove useEffects
* keep 'All' tag after its added
* fix extra newline in code output
* add comment
* Merge remote-tracking branch 'origin/main' into s2-docs-testing-followup-high-priority
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Co-authored-by: Daniel Lu <dl1644@gmail.com>
