Here's the problem with "for profit" healthcare. It's in the doctor's best interest to KEEP YOU SICK. Putting the question aside as to whether a public or private enterprise runs more efficiently (clearly private enterprise does) the real question here is one of "conflict of interest". Where is the incentive for the profit making enterprise to heal people? Because if they succeed, they will then go out of business. In the US, where most people have health insurance (most meaning 84%), you would think that this would balance things out. After all, they are profit making enterprises too. And it would be in their best interest to have payers who do not get sick. So..... they reject certain people for coverage. They find ways to wiggle out of paying claims too.
I was declined coverage once. At the age of 29. 29! Why? Hard to say, but the broker had to coach me then on my answers to find another insurer. Basically, I had to lie. No I never drink alcoholic beverages. No I haven't smoked in the past year. Etc. I imagine if I had truly fallen ill once approved, the insurance company would have done their level best to find out if I had lied on the original form. Aha! We found this picture of you with a glass of wine in your hand from Christmas posted on the internet! You're on the hook for that chemo. Claim denied.
When coverage is universal and "not for profit", the incentives change. Keeping the population healthy becomes the only one.