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Not sure how it's the root, but anyway...
Ultimately the US is one of the most responsive systems in the world. IMO that is a factor in this scenario, as are bed numbers.
Cost of delivery, suicide inducing bureaucracy, inequity of outcome are great for our collective schadenfreude but it doesn't mean that there aren't areas where the US is likely to be better placed that the UK. A much bigger advantage a national health service has over private is the ability to roll out an identical approach.
This is the root of the greatest fallacy about the American Healthcare system, and Capitalism in general; you have the choice to appoint whichever 'health provider' you like within a competitive market, therefore quality must be ensured by that dynamic. Almost without fail, it's the exact opposite.