There is a basic misconception here, which colors the answer: there is no American political system. There are, at any given point, over a million unique, distinct, different political systems operating in America. In many of these systems, referendum is indeed used, so let’s focus where it isn’t.

Some states, not all, allow for referendum petition ballots. Most localities do. The federal government, however, chooses not to (they could, and in theory they do for the senate finally (the root of the 17th has the same basic argument)). This reason is absolutely historical - how would it be done? In an era when we expected to get competing presidential and senate and house member results (and did), had no fast communication (and waited for results for weeks as such), and had a vastly different definition of who would matter (until the fourteenth, and frankly until the 1960s jus singinius change, there was no standard of what a citizen was).

This could be changed now, but frankly, we can see why it may not be a good idea. Inertia, impossibility, and representative instead of direct concepts, are why we don’t.