I've had experience with this both professionally and personally. Here in California, many attorneys are having patients use their health insurance 1st (to get the contractual discount) then using their Med Pay 2nd. Bad for the doctor, good for the patient if they have limited Med Pay on their auto policy. Insurance companies are also having people drop their Med Pay entirely if they have health insurance. (Med Pay is such inexpensive insurance I would encourage people to get as much as they can afford.)
If it's a 3rd party claim (the other person is at fault and their insurance will pay at the end of the claim) the doctor should bill their health insurance if they can so that they're paid in a timely manner. Lots of 3rd party claims don't pay the doctor, they pay the patient, then the patient doesn't pay the doctor!
This may seem rambling, and ultimately you have to bill who the doctor tells you to, but as a biller you need to be aware of the many scenarios there can be. The doctor deserves to be paid for his or her services, and the patient should get their care, so get all the insurance info you can so somebody pays!
Dina