The administration of the US government has no elements of direct democracy (that I can think of). It is a republic with democratically elected representatives.
“A republic with democratically elected representatives” (like “a monarchy in which practical authority is durably assigned to a body of democratically elected representatives”) is an example of a representative democracy, which is not only a kind of democracy, it's by far the most common kind of democracy.
We've entered pretty pedantic territory now IMO, but I think it's much more useful to consider the US government itself a republic because that's how its day to day operation is conducted. Thinking of it as a democracy leads to questions like the thread starter, for instance about un-democratic committee and leadership structures, which are actually non sequiturs because it's not a democratic system.