feat: replace custom search TextField with Flutter SearchBar widget

SearchBar is always visible in the AppBar bottom slot — no toggle needed.
Removed _isSearching, manual debounce timer, and slide animation.
SearchController listener clears results when text is emptied.
Updated E2E and widget tests for the new widget tree.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

next.md

@@ -17,30 +17,3 @@ Git repo should not contain unknown files.
 Then commit.
 
 ## Tasks
-
-How can I edit Sieve Scripts? Afaik this feature was added.
-
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-
-Replace the custom TextField-in-AppBar search implementation in
-lib/ui/screens/email_list_screen.dart with Flutter's built-in SearchBar / SearchAnchor widget
-(Flutter 3.x).
-
-Goals:
-
-Remove the _isSearching bool, the _searchController, the slide animation, and the manual Timer
-debounce
-
-Use SearchAnchor + SearchBar to drive the _searchQuery state that filters the email list
-
-Keep the existing filter logic untouched — just replace the input mechanism
-
-The search bar should live in the AppBar area; if SearchAnchor doesn't fit cleanly there, use
-SearchBar standalone with onChanged (no debounce needed — let the user control submit, or accept
-instant filter)
-
-Preserve all existing AppBar actions (compose, sync, settings) when search is not active
-
-Update or remove any tests in integration_test/ that relied on the old search widget tree
-
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