I'm not so sure that is a sensible inference to draw. Government "interference" in healthcare in many countries where citizens both have greater access and report more satisfaction with the healthcare service they receive.
Also, free markets really don't have much to say for people who don't have the money to trade in that market, and in the healthcare space, denying treatment to people just because they can't afford treatment can cause some massive negative externalities. Just imagine how bad it would be if there was a massive pandemic and only the rich could afford vaccines; there would be so many unvaccinated hosts for that virus that new variants would keep emerging and significantly harm the quality of life for everyone.
Also, market efficiencies are largely a product of the ability of the consumer to compare the price and quality of different services. When the government is the consumer, it A) can afford to allocate this task to a few experts (rather than demand the entire population learn how to evaluate the cost and efficacy of medical treatments), B) can use its monopsony power to negotiate prices, and C) can use its treatment and outcome data to identify opportunities to do preventative maintenance rather than the more expensive (in both life and money) emergency care.
I'm more sympathetic to free markets than most, but healthcare is arguably the sector where free markets would produce the worst outcomes.