You aren't the only one whose work has involved medical insurance.
From my experience, it was quite uncommon for the practitioner themselves to be preparing and sending an invoice to the insurance company. Typically they would pass this off to the billing department, though in smaller places like a dental office I've seen the 'billing department' just be the front desk. It's a touch misleading to equate the billing hospital with the doctor as if penning invoices is a shared responsibility between the two with each taking roughly equal part.
In any case, it's irrelevant. I'm not here arguing minutia about on which continent the responsibility specifically failed, nor whatever individual made an error. The system failed these people, even if the end result was only a momentary heart rate increase from seeing a medical bill until a phone call resolved the issue.
If my underlying point remains unclear, I doubt I can clarify further.