We have a ton of public intervention in health, it’s just not very well done. The US government spends more per capita on health care than other developed countries, while covering only a fraction of the population. We’ve had universal health care for almost forty years, it’s just bad and nobody wants to call it that. A law called EMTALA says hospital emergency rooms can’t turn people away for being unable to pay, so anyone can get care, just in a very inefficient way. Properly covering everybody is Socialism(tm) and therefore isn’t in the cards.