The right often mistakenly identifies republicanism at the founding as a conscious distinction from democracy. For the founders it was a distinction from monarchy and aristocracy.  The idea of an unfiltered or unrestrained democracy was too unserious to consider. Republicanism was just a prudent and limited form of democracy.

Republicanism was transformed by Madison from a form previously considered suitable only for a small homogeneous nation to the extended republic of the United States; a diversified and expansive nation that would avoid aristocracy as factions offset factions. The goal of a republic to represent the people and avoid aristocracy remained even if the geographic restraint was dismissed.

In The New Republicanism Dan McLaughlin in National Review noted that republicanism was only one facet of the new government:

There are four basic pillars of the American way: Our system is democratic because the government answers to the people, republican because we have no king, liberal because we recognize civil and economic liberty, and constitutional because the rules are written down and bind the government. One can add to that list the three key and interrelated structural features of American governance: It is federal, dividing power vertically between different levels of government; it is separated, dividing power horizontally between different branches of government with distinct functions; and it is deliberative, because the vertical and horizontal divisions and the existence of a written constitution combine to make it difficult to produce abrupt and sweeping change.

Republicanism reflected anti aristocracy in its fight against slavery, and in its early Progressive fight against the rise of an economic aristocracy.  Teddy Roosevelt, our first Progressive President, was a Republican.