I think the analysis for why the US system is so screwed is pretty simple.
Since healthcare is a for-profit business, the companies which offer healthcare has as their primary interest to make a profit.
That means (a) overcharging (b) getting rid of people who cost money (those who are ill).
Those two things cannot be fixed by the free market because they are happening because of the free market; the only way to fundamentally change this is by imposing rules, one way or the other.