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This seems to work and no change in the unit tests was needed
after removing the clutter of how StateChecking::DoHandle works.

High level, the state transitions kNew->kChecking and kChecking->whatever
don't need to count callbacks. Instead, the code relies on the sequence
provided by the task runner.

This streamlines the execution path of UpdateEngine::DoUpdateCheck since
it makes clear that it can't be invoked multiple times during
a single call to UpdateEngine::Update.

Bug: 1146864
Change-Id: I116e0f68b6668856aa3c3564ed4f51bf7efd4026
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2523788Reviewed-by: Joshua Pawlicki <waffles@chromium.org>
Commit-Queue: Sorin Jianu <sorin@chromium.org>
Cr-Commit-Position: refs/heads/master@{#825456}