It's hard to compare the US to other developed countries, because the systems that exist in other developed countries could not exist as they do today without the US. While there are high barriers to trade (both artificial and natural - you can't fly across the Atlantic for emergency treatment, for example), you have to look at the global market to get a sense of how it works.
For example: today, companies that engage in medical and pharmaceutical R&D receive most of their revenues from the US. This is true even of pharmaceutical companies based in Europe: the US is their primary source of funding for R&D. (Similarly, 50% of all R&D in the entire world takes place in the US).
Any attempt to bring costs down in the US - whether single-payer or market-based - would result in higher costs for drugs and other supplies to the rest of the developed world[0]. Put another way, countries like Denmark and Canada and the UK have the appearance of negotiating leverage only because they're such a tiny market[1] compared to the US - and, crucially, a completely segregated market as well. So even if pharmaceutical companies end up selling drugs at very low rates there, they can easily make up the difference in revenue by increasing prices in the US[2].
The US doesn't even have to impose its own price controls to bring costs down - all they have to do is permit drug reimportation from other countries which honor US drug patents (like Canada). This would immediately cause drug prices in the US to drop (and, at equilibrium, drug prices in other countries to rise). There actually was a bill introduced in the Senate last year to do this, but it was voted down.
[0] It wouldn't affect countries like India or China, because they already don't honor most relevant pharmaceutical patents.
[1] Again, by revenue, not necessarily by population
[2] This doesn't actually affect their sales much, because sales are relatively inelastic, thanks to insurance - the costs are borne collectively by the population.