It's the same deal as the tech shuttles: in places where public transit is good (NYC, Chicago, Boston) the tech companies don't run shuttles because there's no need.
> What do the US school buses and their drivers do for the rest of the day and on weekends? Just sit idle?
Yes. The drivers typically work a "split shift" with time off in the middle of the day, and the buses sit empty both then and on weekends.