Have you tried to find a new GP for an adult with private insurance recently? Many areas have a shortage in general. This has little to do with insurance coverage, and everything to do with the bad incentives of the medical profession. Who would become a GP with your MD, when you can make twice as much money as a specialist instead? It's like getting a CS degree from Stanford and deciding to become a sysadmin at a high school with it, instead of becoming a SWE.

Conversely, I'm surprised you found a dentist and a mental health provider that takes any insurance at all, and didn't have a months-long wait list. I've almost always had to self pay for both...