There are many elements of Athenian democracy that can be seen in American democracy nowadays. For example, people can vote on the topics that concern them the most, much like Athenian democracy allowed Athenian people to do. Moreover, Athenian people, like American people, knew that participation in politics was both a right and a responsibility of citizens.
These ideas have endured for so many years because they resonate with citizens. As American culture is, to a large extent, a legacy of Europe, Americans are very familiar with the ideas of classical Greece, and deeply believe in concepts such as democracy, citizenship and political participation. Even if some of the ideas are not objectively better or worse than those of other systems, they have most likely endured because of the cultural legacy that allowed for the creation of this nation.