My experience with how this works is they send the whole thing to insurance. Then insurance determines how much they will pay and notify you. They then send payment to the hospital. The hospital receives it and applies it your account and then figures out what you owe and adds it your account. All of those steps can take time so it may still be forthcoming.

It's also possible your balance is not with the hospital at all but rather some third party provider. Often the doctor, the anesthesiologist etc.. bill separately and you could owe them and not the hospital