^^ this is correct.
The only way taking patients from the same insurance panel and charge them cash would work is the provider would work at two different facilities where one takes insurance and one does not. This is common and legal. And one might argue more than ethical, because if it's illegal even fewer people would work in public/non-profit psychiatry. Typical scenario is a hospital/ER attending who has a part time cash private practice. When you contract with an insurance company, you agree to accept rates for all their billing codes, not just the inpatient codes.
However, due to quirks of the US healthcare system, especially for large facilities, billing codes are often separately negotiated with individual insurance companies, except for Medicare. For example, 99213 charged by an internal medicine department may be differently reimbursed compared to one charged by a plastic surgery department, and there's absolutely no uniformity with more expensive codes (i.e. spine surgery done in two different departments or even two different hospitals under the same academic affiliation). This is not as applicable with psychiatry, as typically we don't get reimbursed much (differently to our sister departments), and very often academic/non-profit jobs are funded by public block grants, not pay per visit, or managed in an ACO (i.e. Kaiser, VA, etc). For solo practices, you typically have little leverage to negotiate with panels, but interestingly my colleague was able to get a 10% raise out of Insurance X because there are so few psychiatrists in the area who take insurance X, so this rule is not hard and fast. I tried to negotiate with insurance Y in order to opening a new clinic, but they weren't interested. I live in a saturated area, but if you are one of the few providers in a smaller homogenous area the game is very different. The local provider group is essentially an EAP (say of a manufacturing plant), in that case you might get more bang for your buck if you go directly to the payer (i.e. the executive of the plant). In that case negotiation might be easier even if you are a solo/small group, and you see a lot of the plant employees and can get them to help you complain. However, likely the provider would also be able to separately build a practice outside of this and take pure cash.
This is perhaps too much information, but I suspect this is not the case in OP's description. More likely what is described is someone who takes ONE specific insurance, and whenever a new patient shows up and has that insurance, he will turn that patient away. But his cash bottomline isn't hurt because he has enough of people from other insurances coming in. This is extremely common both for employed and solo psychiatrists and full disclosure I even do it myself (though I'm closing out participation in this insurance in a few months). The insurance is typically an EAP with a limited patient stream that pays somewhat better than the prevailing rates if you try to directly panel, but nowhere close to the cash rate. I have a feeling that insurance companies know that people who try to directly panel are low demand providers and therefore lowball.
I suspect that if the healthcare system goes single payer, cash pay doctors will convert to the current direct care model prevalent in primary care (i.e. insurance + membership fee) with little impact to the financial bottomline and likely an increase in demand, unless such a model is legislatively prohibited (a true price control). In which case you will see fairly severe shortages (especially of severe time consuming cases) where you cannot find a psychiatrist even if you have money.
