CrOS Settings: Fix settings search field focus loss issue

The value of `showingSearch` is only the indicator of showing the search
field or not when it's at narrow status. The change of showingSearch's
value shouldn't manually invoke the focus / blur on `searchInput`. Thus,
this CL removes the function `showingSearchChanged_`.

The only place we need to manually call focus is when search icon is
clicked in the narrow state since the searchInput is hidden at the
moment. The only place we need to manually call blur is when `escape`
key is pressed.

Screen records:
 . before fix:
https://screencast.googleplex.com/cast/NTE1NjA4NTc5NDA3ODcyMHwyZmM5N2YzMi1lMQ
 . after fix:
https://screencast.googleplex.com/cast/NjM2MTc5OTUzMTE2Nzc0NHwwYzAyMzI5Ni1mZQ

Bug: b/316019447
Change-Id: Iafda5ad9f0d1e0ebe57e29c34b3e5afa22930dc3
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5190830
Reviewed-by: Wes Okuhara <wesokuhara@google.com>
Reviewed-by: John Lee <johntlee@chromium.org>
Commit-Queue: Connie Xu <conniekxu@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1249719}