I would certainly want a choice when I become a Senior and need important health care.
Living  in Australia about 15-20 years ago , I was happy on a couple of occasions to forego a summer vacation by paying to get immediate health care for my spouse by using private services . It was NOT an emergency and the alternative was to wait a couple of months.  And yes, we were paying a  1.5% premium tax for Public Health care and also paying optional private health insurance.
We are deceiving ourselves when we think Publicly supported health care ONLY is sustainable with increasing demand, sicker and older people. People are just living longer. That is a fact.! 
On the other hand, a "free for all" private system will be disadvantageous to the poor or those who cannot afford it and that is very important too!
The REALITY of today suggests a mix of the Public and some Private care with clearly defined parameters and guidelines (so patients know who is paying for what) might be the way to go!