I am sorry this has been your experience.  My country has a twin system:  free socialized healthcare for anyone who wants it plus private health care for anyone that wants to pay for it. Many top doctors work part time in both systems, they work private for the money a few days a week and then work public a few days a week so that they get the most challenging cases and can also teach med students.

If you are critically ill, you will be moved out of the private hospital and into a public tertiary hospital, by government owned medivac helicopter if necessary. These public tertiary hospitals are where all the serious research and subspecialty training happens and people with super rare and complex issues are treated.  Private healthcare is more about getting a fancy private suite in your hospital with cut flowers if you are seriously but not critically ill.  

We spend far less than the United States on health care on a per capita basis, but we destroy the US in terms of positive patient outcomes.  This is because the US system is horribly inefficient as has been well documented.  Our government is far more efficient at delivering serious health care than the private sector.  We pay a lot of tax but strangely it seems no more than high income Americans. Maybe we pay less after taking into account your State and municipal taxes.

When I hear Americans complain about free health care, I think it has nothing to do with patient outcomes and is really because they want to cut their taxes.  Maybe if you spent a little less on defence spending instead, then you could pay less taxes.  Do you guys really need a $4 billion Zumwalt class warship with artillery shells that cost $1 million per round?  I think not.  This is where your tax dollar is being pisssed away.