[headless] Use TaskScheduler APIs for posting former FILE thread tasks.

FILE thread is going away. Patch is part of TaskScheduler migration.

Bug: 546953
Change-Id: I1a9d8d63e1a89e1496038e247dda707e15f93418
Reviewed-on: https://chromium-review.googlesource.com/541303
Reviewed-by: Alex Clarke <alexclarke@chromium.org>
Commit-Queue: Eric Seckler <eseckler@chromium.org>
Cr-Commit-Position: refs/heads/master@{#481189}
9 files changed