[iOS] Add search field for the bookmark folder chooser

When the user wants to move a bookmark to a different folder, there is
no search field, making it hard to find the right folder.
This CL adds such a field, still keeping the distinction between local
and sync bookmarks.

https://screencast.googleplex.com/cast/NjQzMDk2ODIyMjA1NjQ0OHwwOTg5NzM1Mi01ZQ

Bug: 406164642
Change-Id: I73885130753ede1f39c28389a50b452876a2211c
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6387705
Reviewed-by: Olivier Robin <olivierrobin@chromium.org>
Commit-Queue: Gauthier Ambard <gambard@chromium.org>
Auto-Submit: Gauthier Ambard <gambard@chromium.org>
Reviewed-by: Louis Romero <lpromero@google.com>
Cr-Commit-Position: refs/heads/main@{#1450445}