Not really in my experience. You get to choose between providers in your insurance network, but I've lost count of the number of times my preferred provider wasn't in my network. On top of that, the insurance company has to sign off on many diagnostic tests even when my doctor wants to do one.
I'm experiencing this right now with chronic pain associated with an old injury. My doctor wants to do an arthrogram, but my insurance won't approve it without doing a full course of physical therapy first and doing any kind of motion with this injury is excruciatingly painful. I have to literally suffer through 12 weeks of PT in order to get the imaging I need and I have what many would consider to be great insurance. I don't really have any (good) choices here. I can pay for the imaging out of pocket or continue to suffer. This is a uniquely American problem.