Anesthesiologist doesn’t control how long the surgery takes.
I have been a victim of surprise billing from an Anesthesiologist.
My solution for surprise billings is to require baseball arbitration between the provider and the insurer, and require the insurer to not penalize the patient for out of network care. Other cure would be require the in-network provider who contracted with someone out of network to eat the difference and leave the patient out of it entirely, coupled with antitrust action against the practice groups where appropriate.
The Pure Free Marketers here will flame me (or should if they are true to their beliefs) but some folks need regulating.
They were trying to avoid what you suggested. Putting anesthesia and patient on notice due to everything being surprise billing from the anesthesiologist.