“We are a constitutional republic, not a democracy” sets up a false choice, and I hear it often. A republic elects its leaders instead of inheriting them. A representative democracy lets citizens choose those leaders. A constitution limits what government may do. The United States is all three at once.
Political science counts representative systems as democracies. The slogan usually redefines democracy as direct mob rule, which is not what the word means.
The framers did build features that limit majorities. They include the Bill of Rights, equal Senate seats for every state, the Electoral College, and checks among the branches. None of that is in dispute. Why they built them is under Slow by Design. Those features limit majority rule. They do not replace popular sovereignty as the principle underneath.
When policy often goes against national majorities, that is the limits being worked, not the system working as intended. The gap is contested, not settled. Efforts to widen voting access, curb gerrymandering, reform the Electoral College, and revisit the filibuster share one aim. They would bring practice back in line with the democratic ideal. The slogan is most often heard in defense of the current arrangement. My position is that the arrangement is a disputed balance worth adjusting, and that majority rule remains the engine even with guardrails on it.