Private operators in a public health care system can help improve the overall efficiency of the system, as long as continuous care is taken in defining their role.
In Denmark treatments are guaranteed by law, and if the public system can't cope the government has to pay for private treatment, this includes in theory treatment overseas.
In the US a system of public hospitals in direct competition with private hospitals does not seem possible.
Too much ideology to overcome, and even if it was government lacks the strength to kick the private operators back in place when needed.