These origins are important. If there’s substance behind “We’re a republic, not a democracy,” it’s not as a description of American government. There’s really no difference, in the present, between a “republic” and a “democracy”: Both connote systems of representation in which sovereignty and authority derive from the public at large.
The point of the slogan isn’t to describe who we are, but to claim and co-opt the founding for right-wing politics — to naturalize political inequality and make it the proper order of things. What lies behind that quip, in other words, is an impulse against democratic representation. It is part and parcel of the drive to make American government a closed domain for a select, privileged few.