They need a device to build tension. If someone walks onstage and very calmly solves all the problems, you've just killed your play. Easy way to raise the stakes is to lessen the time available to the characters. And since we tell people the time anyways... ;-)
I guess I see it as fairly logical that, being backstage in one medium, we stay "backstage" when transferred to another medium. Not that I wouldn't appreciate a good portrayal of a stage manager. There's a book called Broken For You by Stephanie Kallos, and the main character is a stage manager. Until she becomes a...well, spoilers. But the book describes her job in a positive and knowledgeable manner.