If you have something rare or complex, it's probably actually better to be treated by the public system rather than the private system, simply because they tend to have the more specialist knowledge, but you need to either understand the system, be a senior civil servant or have something that they are prepared to treat quickly as otherwise there is a long long wait.
I was diagnosed with a rare and serious but not immediately life threatening condition earlier this year which needed surgery. The surgeon I went to see under the private system, is one the best known in his field and was recommended by my family doctor. However, once I had been diagnosed, he advised me that it would be better that I was treated by a Professor of Surgery at QMH. I was referred as a private patient and have been treated quickly, with the latest techniques (I have a feeling there may be photos of my procedure doing the rounds at various conferences). The care (and the food) is the same as a public patient would have had, the speed of appointment, the private room (though not nearly as plush as in the private hospitals) and not to mention the cost have been very different.
One downside of being treated as a private patient in a public hospital rather than a private hospital is that I have to pay for my treatment first and then obtain reimbursement through my insurer, rather than the insurer being directly billed. In my case, this involved having to pay a deposit of $165,000 upon admission which had to be settled by either credit card or cash.