[Offline Pages] Close race condition of multiple StartProcessing callers.
Introduces new bool to track the starting/picking phase so that we avoid concurrent pickers
(and not overwrite the schedule_callback) in order to also avoid the DCHECK that guards concurrent
SendRequestToOffliner calls.
BUG=
643455
Committed:
https://crrev.com/81356643cfe6ffe145741fb4b8fef53a3db86a50
Cr-Commit-Position: refs/heads/master@{#416957}