“We live in a republic, not a democracy” is a false binary. The United States is a representative democracy in the form of a republic. It’s both.
It’s like saying you’re not in a car, you’re on a road. You may be driving yourself (direct democracy) or choose someone who’s going your way (representative democracy). Or you may be tied up in the trunk…but hey, you’re still on the road, right?
When someone says we live in a republic, not a democracy, they’re downplaying the importance of representation in favor of structure. China and Russia are structured as republics too.
At best, it’s the Dunning-Krueger approach to being pedantic: look at the first definition in the dictionary, think you know enough to pull a “Gotcha!!!” on someone, and stop reading before you find the second, third and fourth definitions.
At worst, they’re actively telling you that you don’t deserve and shouldn’t expect representation in government. And considering that it’s usually said in response to someone complaining about a lack of representation, well… 🤔