Look, I feel your pain. I agree with you. I'm sorry you don't like it. But that's not how the system works.

When you have a surgery, you're receiving services from several different entities, each with separate billing procedures. First, we have the hospital. The bill from them is your room charge, including the nursing care, linens, equipment, etc. Then you have your surgeon. They may or may not be an employee of the hospital, and may or may not be in your network. You'll get a separate charge for their services, either way. You have your anesthesiologist. Same deal. Finally, in the event that you had a tumor or something removed, you have lab services. The specimen is sent to them, they analyze, and you get a separate bill.

Surgeons and anesthesiologists have privileges (they are allowed to provide healthcare) at multiple facilities. They will not automatically be an employee at the facility you receive your care at. There are a gazillion different health plans out there, each covering different providers and facilities. The hospital is just providing the venue for your surgery. Why is it their job to check and see whether all of the providers are covered by your insurance? They're only billing you for their portion. They made sure they were in your network. They're not in charge of the other providers.

You know all of that paperwork you sign before you have anything done? It very explicitly explains all of the above. It should never be a surprise, to an educated healthcare consumer.

I'm sorry if this analogy falls flat, but I can't think of a better one. Think about it in terms of a wedding. You get married at a venue. It's your biggest charge. Sometimes, a coordinator is even included. Awesome! They help you hire your other vendors (photography, florist, caterer, etc.) They make sure the vendors are in the right place at the right time, and make sure they follow the rules of the venue. But at the end of the night, you write a check to each of those vendors. Their services are separate from the venue's. If there's a problem, you deal with them directly, you don't go through the venue.