"Single-payer" means one insurance plan for everyone. France, as you know, has multiple plans for its people. So it's not single-payer.
But you are of course correct in France having private doctors and hospitals as well as public hospitals. Speaking as an American, city17 makes the common mistake of thinking that Canada and the UK, the two foreign countries we are most exposed to, are representative of the rest of the developed world. Since both countries have single-payer systems with zero/minimal cost on delivery (and more or less 100% public hospitals in the UK), many Americans think that that's the way all countries' healthcare systems work. They have no idea that a 30% copay is normal in France, or that Germany/Austria/Switzerland/Netherlands all have Obamacare-like mandates to obtain insurance from one of dozens/hundreds of competing plans, or that Australia has a government plan but *strongly* encourages people to get a private plan.