You stated that if the free market worked with healthcare, we'd be using it. But we don't know if it would improve the situation since it's been so long since the market was free in any sense.

Look at Lasik eye surgery for a counterpoint. Yes, this is regulated by requiring licensed practitioners, but it's not controlled by insurance companies, Medicare/Medicaid etc. The cost for this procedure has dropped dramatically, while the procedure itself has improved in the last decade.