The problem is that for every hour that a private pay client takes up, there is one less hour for a client who cannot pay and must rely on said health care system. A private pay client has the resources to use any of multiple providers, an indigent client can only go to the free clinic. I don't see why you should be entitled to use the services of an indigent clinic, whether you pay or not, as you are taking care away from somebody who has no other place to go.

Some T's make this work by having a private practice on top of a job at a clinic, but your T has chosen not to do so.

It is not always desirable to change therapists, but as the cliche goes, there are definitely more fish in the sea.