the hospital should bill your insurance company but not to

send the bill directly to you. The hospital is doing something wrong.

From your post, I believe the hospital is a in network hospital which means it has contract with your insurance company. All the services are paid based on the comtracted price. If there is any balance due to you, your insurance company will tell you how much it is.

You don't need talk to the hospital except tell them go deal with your insurance company. Meanwhile keep your insurance record/benefit discription in case the hospital uses collector.