As best as I have been able to tell the cliche, "America is a republic, not a democracy," originated during the Eisenhower Administration. It may have originated with the John Birch Society. Reactionaries hoped that Eisenhower would reverse the economic reforms of the New Deal. Eisenhower refused to do so because he knew that those reforms had broad popular support.
Reactionaries decided that that did not matter because the United State was not a democracy.
We should not be surprised that the greatly over rated "Founding Fathers" feared democracy. They were rich men. Many owned slaves. They feared progressive taxation. Most were opposed to public schools.
The Founding Fathers were intelligent, and well educated for our time. We should not assume that they had insight into the kind of government we need now.