>UK is effectively a two party system (lib dems are completely useless and UKIP is a meme) and it has the NHS.
The devil lies in the details here. It is not necessarily just the 2-party system in and of itself, but a) its rigidity and b) the parties it contains. The United States has had the same 2-party party system (Republicans and Democrats) since about the Civil War. The United Kingdom's 2-party system (Conservatives and Labour) only goes back to 1922. Secondly, we have to look at the nature of the parties. Neither the Democratic party nor the Republican party have remotely socialist or labor roots, while the Labour Party was explicitly (as its name suggests) founded around labor interests and has promoted socialism (in one form or another) throughout much of its history. Unlike pretty much every European country, the American working class has never had a major party of its own, so pursuing policies that benefit the working class, like an expanded welfare state, is much harder.
The main reason I suspect that the 2-party system is more rigid in the US than in the UK is that the US has a presidential system (i.e. leader elected directly* by voters) while the UK has a parliamentary system (leader elected by legislature). The first-past-the-post electoral system (which both the US and the UK have) is well known for creating 2-party systems due to the spoiler effect. However, which parties will perform best will vary greatly from one place to another. If you have a district that is heavily dominated by one party, a new party can run there without having to worry to much about the spoiler effect, as it will be unlikely to cause a spoiler effect. You might end up with no party having a majority at the national level, but this isn't that big of a deal in a parliamentary system since you can form coalitions. However, in a presidential system, you don't just have elections in individual local governments, states, and congressional districts, but also at the national level. The spoiler effect is almost always unavoidable for a third party in presidential elections, so presidential elections tend to fuck them over. They are either forced to merge with one of the major parties (like the Populist Party in 1896) or fade away into oblivion (like the Socialist Party) as a result. I hope this makes sense.
>The root of America's problems runs much deeper than "they don't have a handful of Green Party congressmen instead of Democrats."
What would you say are "the root of America's problems"?
>You could also just as easily say that America would get things done much faster with just one party.
Well that's probably true. The question then is, would those things actually be good for the American people? I mean, Nazi Germany was a one-party state, but I doubt you would exactly use them as a role model for how a society should be run.
*Technically the president is elected by the Electoral College rather than voters, but electors typically just vote according to how the plurality of voters in their state voted