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Since base::ServiceDirectoryClient is being deprecated, remove the last
of the references to this class and replace with sys::ServiceDirectory.

Some references to the class are still in startup_context.h/cc because
there are external dependencies.

Testing: CQ.

Bug: 974072
Change-Id: I55f6166792c86459fac7ad46d5cf733060394af9
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1894154
Commit-Queue: Sharon Yang <yangsharon@chromium.org>
Reviewed-by: Wez <wez@chromium.org>
Cr-Commit-Position: refs/heads/master@{#712302}