This doesn't align with real-world statistics. Public options like the UK's NHS provide a higher quality of care, for a lower cost, and they're quicker in emergencies.
This isn't just the NHS, however. Virtually every developed nation's public option, because pretty much all but the US have them, outpaces the US in virtually any metric you can choose.
Not only do we pay much, much more, but we also get lower quality care. We also get slower care. We also get more extreme care.
The core issue with the private sector is they have absolutely zero incentive to provide good care. If they're smart, they should provide suboptimal care and unnecessary care - that way they can get more money in the long run.
The "free market" is not the magic bandaid to fix everything on planet Earth. You HAVE to think about incentives. What will the free market actually do here? Even a cursory, naive analysis will show the free market should promote sickness because sick people don't have options. This doesn't delve into insurance, which quite literally has an incentive to not give out treatment.