[CrOS Settings] Add synchronous support for SearchHandler

(1) Change internal implementation to be synchronous, now that
    LocalSearchService exposes a synchronous API.
(2) Add a synchronous public API for SearchHandler in anticipation for
    use by in-process clients (e.g., upcoming Launcher usage).

Bug: 1063505, 1059099
Change-Id: If1110a4f43158ca17b387d766dac58cc87ba08f3
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2130681
Commit-Queue: Kyle Horimoto <khorimoto@chromium.org>
Reviewed-by: James Cook <jamescook@chromium.org>
Cr-Commit-Position: refs/heads/master@{#755197}