[Extensions] Clean up dependencies in extensions/browser/api. (1a)

This a reland of f3df55ae2de6e4bb1d91bc9115877faf550a2378. It adds
an explicit public_dep to the automationInternal API that's needed
for a fuzzer test.

This is the first in a series of CLS to clean up the dependencies in
the various API implementations.

Bug: 40593486
Change-Id: I00bb1fdbab85778caa7eae228e048332610626c1
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6090029
Commit-Queue: Emilia Paz <emiliapaz@chromium.org>
Reviewed-by: Emilia Paz <emiliapaz@chromium.org>
Auto-Submit: David Bertoni <dbertoni@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1395196}