Not it's not exactly true. This figure is for exclusively office based private practice psychiatrists, which make up fewer than 10% of all psychiatrists in the US. Most psychiatrists who do private practice have another job either as their primary or secondary to diversity their practice, get benefits (e.g. pension, health insurance), or as another revenue stream (most psychiatrists in most parts of the country could not expect to fill up a cash practice). This represents about 1/3 of psychiatrists (who do pp and another job as well). Psychiatrists work in many different settings including psychiatric hospitals, general hospitals, prisons and jails, VAs, academic medical centers and medical schools, nursing homes, schools, colleges, community mental health centers, and HMOs like Kaiser.
Psychiatrists who choose to go into office based private practice don't tend do it for the money (you could in many circumstances make more money taking insurance) but for autonomy, because they wish to have a psychotherapy practice, want to choose their own hours, don't want to deal with the hassles of billing or insurance companies setting limits on treatment etc. Most patients outside of specific major metropolitan areas expect to be able to use their insurance for their mental health care. Also in some parts of the country pays much better than others (for example Oregon) so you see more people taking insurance, though yes it will typically be limited to certain insurances that are less hassle and offer decent renumeration.
The average high volume med management practice that takes insurance would make more money than the average psychotherapy based cash practice.
So it's about 5% of psychiatrists who do only office based pp and take only cash (though there is another 15% who do some pp and don't take cash). The bigger issue is that most pp psychiatrists don't take medicaid or medicare, and there are many insurances that also wont be accepted because they are so terrible. Most of the Obama care plans are pretty bad when it comes to mental health but will get you into a CMHC - which in my areas have become inundated with referrals and can't cope.