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Followup to https://crrev.com/c/1989533

Sync supports delaying the engine startup by 10 seconds. This is meant
for avoiding contention during browser startup.
Before this CL, the 10-second delay would also apply in one other
(rare) situation, for no good reason other than some tests depending on
this behavior. Two of these tests were recently fixed (in
crrev.com/c/2339986), and the one remaining one is fixed in this CL.
With that, the delayed startup can now go away!

Bug: 1035874
Change-Id: I865d0bc3caac4d8f0712489ccbad526f912ccdce
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2343036Reviewed-by: Maksim Moskvitin <mmoskvitin@google.com>
Commit-Queue: Marc Treib <treib@chromium.org>
Cr-Commit-Position: refs/heads/master@{#796819}