There seems to be a lot of misunderstanding of American politics by Europeans. You're not supposed to vote directly for leaders; it's by design.
Even a pure democracy is natural rigged to defer to tyranny of the majority. Voters would simply stop voting for taxes until government was bankrupted out of existence. So yes, in that sense, our representitive democracy is working just fine. The general public is notoriously bad at understanding geopolitics, foreign policy, or the economy. I, for one, do not want a majority of Joe Sixpacks in a room deciding if we go to war or not.
As for our two party system, there are a lot of examples of multi-party systems that are a complete failure (Italy). The prime minister needs a majority in both the Senate and House of Deputies which is nearly impossible, so nothing gets done. Obviously you need some balance between a single party system and a pure democracy, go figure. You'd be surprised how effective a benevolent dictatorship can be (e.g. Pax Romana), so our founders tried their best to come up with a reasonable compromise.
That said, corporations do tend to trip up the system, I'll be honest. The US has lobbying groups and campaign finance problems, but those things too have a valid place in government. A business owner should be able to petition the government! Sometimes the corporation is an expert in a subject that the public is not. There should be ways to reconcile these differences but majority rule or restrictions on free speech are not the answer. Regardless, any path should be tread carefully or we risk removing the very safeguards in government that are designed to protect us from ourselves.