The Greek concept of democracy looked nothing like the modern republic. A Greek democracy did not elect legislators: that function belonged to the people assembled as a whole. Nor did it elect ordinary magistrates, who were instead selected by random lot from lists of willing volunteers in a process now called "sortition." The rationale behind sortition relied on the notion that the ordinary business of government was mundane and required no special competence or knowledge aside from the ordinary citizen's share of justice and patriotism. The ancient Greek democracy elected only ambassadors and generals, people in whom authority needed to be concentrated and yet who also had to be competent rather than simply willing.
There was an ancient form of government in between tyranny and democracy: oligarchy. In an oligarchy, a few people were selected to make decisions for the rest. There were in different cities various methods of selecting the oligarchs, but popular election was certainly among them. The downside to regularly scheduled, popular elections, in contrast to the random chance of democratic sortition, was that the expectation of elections generated political parties or factions to attempt to influence or control the outcome of the popular election and thus control the oligarchy. This was, of course, impossible in a democracy, since no election was held: sortition makes partisan politics pointless.
In short, the reason that the word "democracy" is never found in the American Constitution is that America was never intended to be a democracy in the ancient sense, the only sense in which the Founders understood the term. The modern "democracy" of an elected legislature and elected magistrates is, in fact, what the Greeks would have termed an oligarchy.