A big selling point of various universal health care systems is how they're fairer. No one is turned away because they don't have money. Everyone gets the care they need in a timely manner.
Dafydd ab Hugh of Big Lizards has a revealing post about his father's experience with Japan's oh-so-fair national health system. It's a tale of private insurance for the well off, peddling influence, bribery, and work that nurses and doctors won't do.
If you thought the French letting its elderly die because no one wanted to cut short their August vacation was bad, just look at this glimpse of trying to get timely, quality health care during the busy New Year's holiday.
I'm going to be honest. There will always be a problem with more demand for health care than the system can handle. And I'd much prefer a system where people with money have an advantage than one where people related to hospital executives or prominent politicians do.