Let me rephrase. It is evidence, but it is not sufficient evidence upon which to base your conclusion. Magic is irrelevant. Factors affecting the chart in the OP include, among other things: pollution, diet, climate, family construct, stress levels, physical activities, water, and so on. To say access to healthcare is what produces the numbers in the chart is an unsupportable leap when many other factors exist. There's a reason scientists set up controls when performing experiments.
So here is what scientists at the Institute of Medicine of the National Academies had to say in 2004:
"Lack of health insurance causes roughly 18,000 unnecessary deaths every year in the United States. Although America leads the world in spending on health care, it is the only wealthy, industrialized nation that does not ensure that all citizens have coverage."
An enormous amount of data links lack of insurance to lack of access to healthcare to poor health in the U.S. Even when you control for other factors, lack of insurance/access is correlated with poor health outcomes.
Just the other day I spoke with a woman with cystic fibrosis who volunteers for a legal hotline for health insurance. She said just recently a 9-year-old's private insurance wouldn't cover a liver transplant, and the law requires 4 weeks of waiting before you can get on public insurance. She died waiting for the coverage to kick in. Conservatives say no one is allowed to die in a hospital in America because they can't afford treatment. They are living in a fantasy world.
The woman told me this kind of thing happens all the time, but it's very difficult to organize because people who are sick and dying, or bereaved, have a hard time taking political action.
Anyway, a LOT more relevant, scientific information is available from the non-partisan Kaiser foundation:
http://www.kff.org/uninsured/upload/7451-05.pdf
The evidence seems overwhelming to me.