Lack of government involvement is not the problem with U.S. healthcare. There is government intervention everywhere. My rule of thumb for determining whether an industry is a free market or not, is that I ask the question, "Can I start a business to solve the problem? Or am I legally prevented or constrained from doing so?" In the U.S. I am legally barred from implementing most of my business ideas that would fix the healthcare system, so I do not consider it a free market.

The problem with U.S. politics in general is that it is fractured. Policy is determined by the interplay of powerful, organized factions. Any government intervention gets captured by these factions, and the policy ends up hurting the general interest rather than making things better.

Here is a comment I made a while ago where I explain how American politics works in practice: http://news.ycombinator.com/item?id=557298