Ensure `background-processor` exits after any sleep future says to

Ensure `background-processor` exits after any sleep future says to

If the user's sleep future passed to an async background processor
only returns true for exiting once and then reverts back to false,
we should exit anyway when we get a chance to. We do to this here
by always ensuring we check the exit flag even when only polling
sleep futures with no intent to (yet) exit. This is utilized in the
tests added in the coming commit(s).