The issue is one of conflict of interest.

No physician should be allowed to be in both public and private practice thus removing the incentive to self-refer.  No physician working in the public system should be able to refer a patient to a private clinic if they receive compensation for doing so.  If they do refer to a private clinic they should at the same time be required to refer to a public clinic so the patient can consider both options.

This particular problem does increase wait times in the public system since the same physician is working in both.

There are many things wrong with our health care system but this particular problem should be easy to fix given the political will to do so.