I would point out that the attending surgeon, the anaesthesiologist, and the hospital all bill the patient separately here for a surgery, in Ontario, Canada. (Most people will never see these bills and it's covered by the public insurance.)
The reason is that the service - the contract - is organized in that way. Between the patient and the anaesthesiologist, between the patient and the surgeon, and between the patient and the hospital. So three separate bills. The doctor and hospital will have their own contract. Legally, and in an important way when it comes to decide who to sue for malpractice, the hospital is not the provider or performer of the surgery. They just provide the venue and equipment; the doctors do the surgery.
The reason is that the service - the contract - is organized in that way. Between the patient and the anaesthesiologist, between the patient and the surgeon, and between the patient and the hospital. So three separate bills. The doctor and hospital will have their own contract. Legally, and in an important way when it comes to decide who to sue for malpractice, the hospital is not the provider or performer of the surgery. They just provide the venue and equipment; the doctors do the surgery.