Old Labour tried that argument in the UK in the 70s, as well as trying to portray the Public Hospitals as subsidizing the rich, so they closed down all the Private wards in the Public Hospitals. 

Sadly, they found out that reality was the mirror image of their ideology, and that the private wards were subsidizing the public in those facilities, and without the income stream going to the hospitals, whole wards and OR suites couldn't be funded, and were shut down, resulting in longer queues.

All the Private facilities I have heard of here in Canada are separately run, and the article lists many. Where are the hospitals that allow private work to push out public patients? Not here in Quebec.