Hospitals can only bill you for (their) services. So, the ER, CT, MRI, X-Ray, or surgery room, hospital room etc they bill you only.
But the problem is that (most) doctors/specialists DON'T work for the hospital, but have only admitting privileges to use them. So the ER DR, the surgeon, or the other most common are radiologists and anethesiologists (separate) bills you get are from them for their services.
See, most of us will use what hospital or doctor is in our network and covered by our insurance. But, if in the ER, or have surgery, can get stuck with these other specialists that are (not) in network, but many are out of network, that your insurance won't pay, or you get stuck with the difference, since they are not in network, nor do you or any (normal) person say, are you covered by my (X) insurance?
That is the flaw in the system, that you get stuck for bills that are not covered by your insurance.