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This replaces the slower, more generic RunAllTasksUntilIdle() and
saves code to boot.

I thought this might make the tests less flaky, but it turned out to
be unrelated to the flakiness I was investigating.  Still seems worth
doing.

Bug: none
Change-Id: I66814c7596bac79f1e5b05356a86fd87984983cd
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2071586
Auto-Submit: Peter Kasting <pkasting@chromium.org>
Commit-Queue: Scott Violet <sky@chromium.org>
Reviewed-by: Scott Violet <sky@chromium.org>
Cr-Commit-Position: refs/heads/master@{#744365}