re: Why do hospitals have their people send separate bills?

Depends on the hospital or the hospital system. Some hospitals employ certain physicians (anesthesia, various surgeon specialties) and you may not receive a separate bill for their services.

On the other hand, some hospital contract work out to local physician groups or practices instead. For instance, a hospital may make arrangments with an orthopedic practices to take so many call days in a month and they may do this with multiple practices to fill that call schedule. As such you are likely going to get a bill from that practice or surgeon if you happen to be serviced by them that day.