I have private health cover. My Oncologist and Surgeon both work in the public health system. The difference it would have made to me for surgery if I chose public for my operation was one week. I have been admitted to both the public and private hospitals for high temps during chemo and the care was exactly the same, perhaps even better in the public hospital, it's just that I didn't have a private room or bathroom.
I am going public with my plastic surgeon for reconstruction too.
I see my surgeon and Oncologist of choice and Medicare pays some of the cost for consultations.
My health fund doesn't cover consultations anyway, so no advantage to having private health cover there.
So you can do it that way, and go to the public hospital for treatment if you like, you just have to check if your surgeon etc will work in the public system.
I think it depends too if you have access to an excellent Cancer Centre at your public hospital. We have one on NSW Central Coast, so going public is a good option.