we have it to elect politicians, first past the post, so it was appropriate.
Indeed, but some people who find themselves in a minority don't accept a majority vote and just keep banging on to rerun it till they do impose their will, Nicola Sturgeon, for example. The referendum the Scots nationalists already had (and lost) was stated very clearly to be a once in a generation vote. People won't take any of them seriously if they can be overthrown a few years later, and then we'd be on a very slippery slope.
All govts make mistakes. The most important thing to me is that they are accountable for their actions, and can be thrown out. The EU has quite a democratic deficit. I don't understand American politics, so don't even hazard on view on that!