This phenomenon does happen. In a number of areas, if you want to find a non IMG psychiatrist who does combined med mgmt and psychotherapy, it's almost exclusively non insurance. People who charge full fee get away with it because people know that the psychiatrists who take insurance generally don't spend a lot of time with you. People who can afford full fee with the help of out-of-network coverage generally want some TLC, and think that the 10 people who do full fee privates all charge $300, therefore, they must be better--and they are, in the sense that they speak crisp English, look dignified, sit in beautiful fancy offices, and give their patients their cell phone numbers, and do pretentious psychodynamic treatment. Not in the sense that the Zoloft they prescribe is gonna be somehow more effective.
Obviously you are very unfamiliar with the complex games of big leagues private practice. You really really need a mentor if you didn't go to one of the few residency programs where you get a lot of familiarities with this just by being in them--and this could mean PAID supervision, which would be SO worth it. You will need an initial investment, and at least a period of time for caseloads to ramp up--which means you might need some part time job to hold you over...this process could take a number of years, during which time you either will be on insurance panels and doing high throughput med mgmt, or work for a clinic, or do inpatient/ER work.