European constitutional monarchies are effectively republics with some state-sponsored celebrities. North Korea is a republic that is effectively a hereditary monarchy. Official labels don't always match reality.

Anyway, my point was that "republic" effectively means "liberal democracy" in this specific context, where someone invokes the Founding Fathers and contrasts it with direct democracy. And when someone talks about present-day democracy, they usually refer to the entire Western system rather than to simple majority rule.