Well, Japan and France also deliver healthcare directly to citizens while Canada is essential fee-for service private with a since fee payor. In other words, in Canada doctors and some diagnostic clinics are are private businesses that charge fixed, negotiated rates to the government for services. In Japan and France, doctors and diagnostics are salaried employees of the state. In France, they have 2x the doctors and pay them 1/2 as much.I had heard that Australia's experiment with mixed care had been disastrous, but never got any details. If anyone has references they want to share, leave it in comments.
By all means, lets convert to that system if you dare, but it will be harder to sell than the current one - its far more "socialist" than what we have.
Also, you may want to look at how the mix of public and private ruined Australia and nearly ruined Britain. There are more countries in the world than France and Japan.
It's also worth re-stating that Canada doesn't have public health care, we have public, non-profit health insurance. This is an important distinction.