They pay out of pocket, then seek reimbursement from the insurance companies themselves.
This takes the burden off the doctor and puts it where it belongs, as an issue to be worked out between a patient and their insurance company.
It used to be the case that you paid the doctor. My father still has a doctor's bill from when he was born that my grandfather paid. The entire cost of delivering a child at that time was $17.50. That was in 1937.