if the anesthesiologist was not a contracted provider of the health insurer, then the balance due is owed by the patient.
when having any kind of medical examination or procedure, the patient needs to verify that ALL of the participants in the procedure are contracted providers of the health insurer.
sometimes, the primary care physician is a contracted provider but the other members of team are not.
this puts an unreasonable burden on the patient who is not in a position to determine the status of all the providers.
IMHO
Anesthesia and certain other things are sometimes not covered or may be “out of network”. When I had my kids some 20 years ago we had BC/BS of Central NY, and they paid everything (I don’t even recall a copay) except the anesthesiologist. I paid them $300 out of my own pocket.