As we answer this question, we must keep in mind that the American rebels were not one monolithic group that all had the same goals. This means that different rebels would have been making different points. Let us look at the two main points that were being made.
One point that was being made was that people ought to be able to rule themselves. The colonists did not like being ruled by people in faraway England or by those appointed by the people in England. They wanted local elites to be treated the same as English elites. That is, they wanted their own elites, rather than elites from England, to govern.
A second point that was being made is that democracy is the proper form of government and that monarchy is a bad system. Some of the colonists were rebelling largely because they felt that all the people should be allowed to govern themselves. They were not trying to ensure that American elites governed. Instead, they wanted Americans of all sorts to be involved in self-government.
Thus, there were at least two points being made. One was a point about self-government while the other was a point about true democracy.