Everybody knows that the United States could easily have been a monarchy or a theocracy, but what if something else had influenced the founding fathers? Might they have founded the United States of Confucianism? Or Machiavellianism?
The Founding Fathers drafted our Constitution and our country as an experiment in applied rationalism. Everywhere else, superstitions and class favoritisms were explicitly spelled out in the law. Those who created the government here decided to try doing without all that, relying instead on clearheaded empiricism. That set of ideas came along at the right time to influence the birth of a nation.
Philosophical systems are created and developed by groups of intellectuals corresponding with one another. By contrast, governments are usually formed by a conqueror or they coalesce out of smaller city-states. The United States became independent in an unusually self-aware way. It provided an opportunity to build a good government from scratch - to learn from previous mistakes. The form of "good government" the founding fathers chose was democracy, because it was the idea that was in vogue at the time. But it could easily have been different.