We have a republic, because that's how it was organized more than two centuries ago, and changing it from within the system is extremely difficult. But many states have elements of direct democracy in form of public initiatives, and they generally tend to be the states that were settled and organized later (e.g. West Coast), and did so with benefit of hindsight, after seeing many decades of how the original system works.

So I'm not sure if it's even meaningful to say that we do reject the idea. The original designers did (mostly; they weren't a hivemind, either), but we aren't them; and they weren't really representative of their entire society, either.