Improve mobile inline search and remote options UX

[tyreseluo]

Summary

• switch mobile room search to an inline flow instead of relying on the desktop-style modal
• add inline remote-search actions (Search People, Search Rooms, Search Spaces) for mobile
• keep remote-search actions visible even when local matches exist (mobile + desktop)
• fix the mobile remote-action buttons rendering as blank placeholders by forcing label sync on active adaptive instances
• keep desktop behavior intact while explicitly syncing adaptive visibility for search entry icons
• add/update the task spec for this work under specs/

Root Cause

The mobile remote-search option buttons were created, but in the AdaptiveView + CachedWidget path the label updates could miss the active instance, so the UI rendered button placeholders without text.

What Changed

• src/home/rooms_sidebar.rs
  • added mobile inline remote-search panel state + actions
  • added remote result rendering and click handling
  • added robust adaptive sync for icon visibility and remote button labels
• src/app.rs
  • show desktop remote-search action buttons even when local results exist
• src/shared/room_filter_input_bar.rs
  • exposed a ref method to control search icon visibility at runtime
• src/home/rooms_list_header.rs
  • exposed a ref method to control desktop search-entry button visibility at runtime
• resources/i18n/en.json, resources/i18n/zh-CN.json
  • improved remote-search action button text with clearer descriptions
  • removed TODO wording from search text
• src/home/search_messages.rs
  • removed literal Search (TODO) fallback text
• specs/task-mobile-inline-search-with-remote-options.spec.md
  • added the implementation spec and acceptance scenarios

Validation

• cargo build