Red flag just means that it is something that calls their attention to an item. That doesn't mean there is anything wrong with that item. Document the doctors you have called trying to get appointments and what their responses were.
This is something that will blow over. It is not a big deal to drive a distance to see a doctor, unless that doctor is famous for writing prescriptions for pain killers or something unusual like that. People very often are loyal to an established physician and don't change if they don't need to see him often.
