Medicare is NOT single payer and even then, there are plenty of
doctors who do not take medicare patients.
It's pretty much single payer with options, and the doctors at the
practice I visit are as money-grubbing as any selection of doctors. In
fact, years ago I asked my doc at the time what he thought about
Medicare and he said he was glad for it, because it brought the practice
patients they would not otherwise see and it generated more ca$h.
The current Medicare reimbursement for an hour of psychotherapy
treatment was about $90 last year, according to my wife, whose
professional assn publishes such data. The private insurance
reimbursement rate averages about the same.