It doesn't matter what seems plausible to you, what matters are the hard facts of how much is spent for what outcome, and as you'll see from those graphs, the evidence is overwhelming that the US spends a lot (both per capita and as % of GDP), and gets less in return.
I thought it obvious, but the first hypothesis from the OP:
I'm not convinced that the government can get a better rate
From the data:
Single payer system countries have far lower costs for similar if not better outcomes.
The other hypothesis:
I think the real reason in Japan might be that costs are low because there is more competition due to more private doctors
From the data:
Japan has dramatically lower costs than the US, while having a radically different system far more similar to 'Medicare for all'. The far more private doctors comment was I think addressing irrational concerns that private healthcare would somehow disappear under medicare for all (instead of just being a lot cheaper). Japan is the proof that doesn't have to be the case.
What are better outcomes ? How are those measured ?
US is still the leader in healthtech, health quality, and health service timing.
Canadians are literally pouncing on new doctors like freshmen on beer, because existing doctors no longer take patients. The UK NHS has waiting times that are measured in months.
The only thing we struggle with is cost. So again, what are better outcomes ?
https://slatestarcodex.com/2017/02/09/considerations-on-cost...
https://www.statista.com/statistics/268826/health-expenditur...
https://en.wikipedia.org/wiki/List_of_countries_by_total_hea...
The US is an outlier, because the US system is broken.
IMO the US healthcare market either needs to be properly regulated, or it should be converted to single-payer (a system which many other countries use with great success) - other developed countries are spending significantly less per capita for better outcomes while still allowing private healthcare for those who want it and can afford it.