If you cared only about the poor and the disabled then we would maintain the status quo. Universal Health Care is a better deal for all Americans, rich, poor, healthy and unhealthy than the current system Countries with Universal health care spend less on administration and more on health care.
You are thinking private sector because it has the profit motive can deliver health care at lower prices and better results. For health care, the laws of economics are against you.
For company to make a profit, they must receive more in premiums than they pay out. To cover the costs of those who cannot live without health care premiums must be high. If health care could attract those who are healthy, their premiums would cover the others. No insurance company can lower premiums enough to attract buyers who are healthy because those in the pool are high-cost patients. The premiums are always high and some will be unable to afford insurance. Without employers paying for health insurance for high and low cost insurees, the market for health insurance would collapse.
The value of UH is spread of risk and cost across many people.
I stole my analysis from the "The Undercover Economist" by Tim Hartford. His analysis based on economic theories in the link below.
Even if you got rid of all government healthcare programs, our problems with healthcare would be the same.
http://nobelprize.org/nobel_prizes/economics/laureates/2001/public.html