The argument is that although the U.S healthcare system - based off of some metric - underperforms compared to a system which makes no claim of being based on a free market, the U.S. healthcare is also not a free market by any means. Lightly regulated it is not.

Also, you combine "works better" and efficiency: This is slightly unheplful, as although (let's say) the NHS is very cost efficient is - however - not anywhere near as impressive when it comes to patient outcomes. I am not well-versed in the topic to provide a well thought out counterexample however I believe that the Swiss system would be a suitable example?

Comparing the deregulation of businesses with guns is fallacious if not ridiculous: However, being British etc, I do share your fears as to healthcare deregulation.