To this citizen of a Constitutional Monarchy (Canada); what PrairieLady2 said seems perfectly reasonable. OTOH, I’m struggling to understand what you said, in response. It seems almost like a non-sequitur. Why do you think representative democracy not compatible with a republic?
Canada’s Head of State is Queen Elizabeth II; but, we operate as a representative democracy — our elected Prime Minister is Head of Government. If we replaced the monarchy, with an elected Head of State, we’d be a republic. A constitutional democratic republic, like, e.g., India is today.
The US is a republic. Representatives are elected by citizens and then those representatives legislate according to their own best judgment. The citizens can - if they wish - exercise various mechanisms for removal from office.
Tthe republic was expressly designed to avoid direct democracy.