Chile is a terrible example of this; it has one of the higher levels of income inequality on that side of the continent and privatized education and health care. Private education in Chile is appalling; wealthy Chileans send their kids to study to the US or EU and the middle class that can't afford education sends them to Argentina or Brazil. Capital is concentrated on a few hands with a strong dependency on basic commodities exports with low added value. It's better than many places, but the GDP figures don't reflect the realities of a substantial amount of its population.

Are you just making stuff up? Chile has a public education system with voucher options, but almost all of the schools are publicly funded with voucher options, and even the private ones get government subisdies. And health care is a mix of public and private (like most countries) and free if you are over 60 or make a low income. So I think you are going to need to elaborate on those points a bit more

This "inequality" stuff is really getting annoying. The poor in Chile are far better off than other countries in the region, but because the rich are also richer somehow inequality is all that matters to some.

In fact, the wikipedia article "Poverty in South America" doesn't even mention Chile http://en.wikipedia.org/wiki/Poverty_in_South_America

If the poor can't afford basic health care then there is something really wrong with an economy.

I'll give you Chile as a counterexample, although it's not as recent as the others.