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Reland "Add thread checking to RunLoop, deprecate MessageLoopRunner. (patchset #4 id:20002 of https://codereview.chromium.org/2537893002/ )"

Reverted in https://codereview.chromium.org/2548883002/
Reason for relanding: turns out something else was causing these failures.
See details in https://bugs.chromium.org/p/chromium/issues/detail?id=670844#c17

TBR=avi@chromium.org,thakis@chromium.org,asargent@chromium.org
BUG=668707

Review-Url: https://codereview.chromium.org/2564943002
Cr-Commit-Position: refs/heads/master@{#437632}