Looking at the situation in America, where the insurance companies are driven by greed and excess and take a huge chunk of the profit while providing inadequite cover for their civizens, i would say that your blind faith in the for-profit free market system is naive, especially when we are talking about peoples health.
1. What the US has is
not even close to a free market due to the extreme regulation of the insurance sector
2. It's not blind faith, its based on actual reasoning about how markets work, and how voluntary interactions work better than initiation of violence so that some people can push their costs onto other people.
3. Yes I acknowledge that people's health is important and that's why it cannot be left to the government, because govt routinely fails (at being efficient) and is open to scamming and gaming of the system. The fact that its important means it should not be left to the government - you can't improve things just by mandating that they should be better.
For more details, see this video:
US Health Care