The rest of the OECD has universal healthcare, but not necessarily fully public healthcare (the model vary considerably.)
Which just means that there are lots of good proven models that the US could adopt.
https://ourworldindata.org/the-link-between-life-expectancy-...
The most recent life expectancy numbers are the hardest to interpret, the US has the only three-year-running negative life expectancy growth in the first world nations - but some argue we should separate healthcare from the suicide numbers that are a big contributors. I don't know that I agree with that as there is a fuzzy line between healthcare and social care efforts.
Age-adjusted all-cause death rate is the better stat for measuring life-saving outcomes.
Child mortality is certainly a useful stat for its purpose, though.
Are there sources of that stat over time and between nations that you would recommend?
http://apps.who.int/gho/data/view.main.1360?lang=en
It probably still supports the point you are trying to make, I just get grumpy about statistic selection.
Even in our own VA system, their negotiated prices are lower for drugs than the Medicare part D program, barred from negotiating (and with large pieces managed by private companies).
I think your skepticism should primarily be directed to wondering if private systems can ever negotiate a fair price in healthcare products and services.