The US already has single payer for a good fraction of the population and we still spend more than other nations on that fraction. The NHS in the UK can decide that certain treatments are not cost effective but in the US we have people raising the cry of "Death panels! Death panels!" at even vague gestures in that direction. And the US had never had a particularly high level of bureaucratic efficiency compared to most countries.
Which isn't to say that single payer wouldn't be an improvement on the current system. Just don't get your hopes up for how much money it'll save.
> The US already has single payer for a good fraction of the population
No, we don't. Medicare is not single payer, though it has (for a subset of the services covered by Medicare) a default public option; it also has private, partially-public-subsidized pland; Medicaid, at least in many states, is not single-payer, either, even at the state level, even before considering overlap with Medicare and other insurance.
Which isn't to say that single payer wouldn't be an improvement on the current system. Just don't get your hopes up for how much money it'll save.