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I don't see how it could work in theory. Let's assume you have a public and a private system and somehow the private system magically performs better than the public system. Then people cut support and funding for the public system and all you have is a private system. But now everyone is stuck with it, so at best you have to have a large body of regulators watching over private companies that are trying to flout the rules as much as possible because that's their profit margin. And this must all unavoidably drive up the price, because of the profit margin and the extra people involved.

And that's all assuming that you can properly regulate the system, which you clearly can't, because the more money it has then the more power it has, and healthcare is a massive industry.