You choose to argue, not deductively from the Constitution that you presumably wish to identify yourself with as I in fact have argued but instead you resort to argument by analogy, a weak form of disputation second only to argument by ad hominem. A vehicle that is a truck is not a vehicle that is a car-that enlightens us in some respect?
Perhaps worse, you are just factually wrong. You certainly can have both, in fact, in a decent political system you must have both.
Our Constitution provides for democracy by representation. People's democracies or People's republics, such as the Soviet Union under Stalin, claim to have a Constitution but have in fact ruthless rule by autocrat. They had a fascinating written Constitution but they were not a constitutional republic anymore than they were a democratic republic or a representative democracy. A Constitution without legitimacy born of democracy, is a meaningless scrap of parchment.
A democracy without a Constitution is mob rule. History is replete with examples aping the French Revolution of democracy without a Constitution and the rule of law. Equally history teaches us of so-called constitutional republics exploiting a written Constitution to cover lawless suppression of democracy.
God chose to assemble a unique collection of selfless patriots who contrived a Constitution that is a waypoint between the autocracy of a monarchy and the terror of mob rule. They succeeded in giving us both a constitutional republic and a representative democratic republic, they gave us both, thank God.
And now you've entered a third form of government into the mix.
Believe as you will.