Yes, the Runner Die is effectively a clock for Gig completion! It's separate from the Deadline clock; once you launch a new Gig, it becomes "active", so the Deadline clock no longer matters.
It's differentiated as a die both for ease of distinguishing it at a glance and because the starting value can vary if you make it a Rush Job. Other than that, you don't roll the Runner Die, and you'll only ever use one per Gig, from whoever you chose as the leader.
As to your other comment — I am still working on a full version, but it's low on my priority list at the moment. Any more feedback or questions will help me with it a lot, though!