All democracies today are representative democracies. The reason why the American founding fathers used strange terminology is as because no representative democracy existed at the time. The system we have today with the idea of separation of power as outline by Montesquieu, was not called democracy back in the day. But in the late 1800 our current concepts of democracy became firmly established. It seems very strange to use a sort of 1700s concept of democracy in the 1970s.

A key problem is that it is simply very confusing and inconsistent with how political systems have worked since at least the 1860s.

There are republics which are not democracies and democracies which are not republics. Thus we must treat democracy as a separate characteristic of a country from a republic .

My native Norway e.g. is a representative democracy and a constitutional monarch. It is not a republic. Our neighbor Finland however is not a monarchy but a republic. However both Finland and Norway are representative democracies, just like the US.