If you have PPO medical insurance, or Medicare, you can choose your doctors. You have a yearly limit to how much you have to pay, regardless of how much the actual treatment cost. I just about dropped dead when I saw my first bill from Mayo, and it was the first of three. If I'd been on the hook for the many tens of thousands they charged, I'd probably only just be paying it off.
If you're in an HMO or limited network of providers, yeah, you're stuck with who they give you. (And they may be great doctors -- we've certainly loved the doctors we've seen at Kaiser.) There's no way you'd want to take on the cost of a different doctor yourself.