Our Senators used to be elected by the state legislatures--a practice that was far superior to what we have today--but are now elected democratically as well.
"Direct democracy" is seen frequently in the guise of state referenda which many states utilize.
The ultra-conservative insistence on making "Republic not Democracy" a continuing issue is mostly straining at gnats.
The only time the argument has been pertinent was to refute the Dems' grumping about "popular election" of the President in the wake of Election 2000. Otherwise, it is a distinction without a difference.
I disagree from a philosophical standpoint, however from a practical point, you might be right, since the average American does not understand the difference. I like the following definition of Democracy: Two wolves and a sheep voting on what to eat for dinner.
I think the distinction is important. I think it is also important to note that many things (constitutional amendments) are not strictly "democratic" in that the majority rules, but a Super Majority (66%+) is needed.