A republic is any government without a monarch. The USSR, NAZI Germany, and Pol Pot’s Cambodia were republics, while Churchill’s Britain was a monarchy.
Democracy is the rule of a majority of the people. France during the 1793-4 reign of terror was both a democracy and a republic, and also a really terrible place.
What is most valuable in our political tradition is the limitation of government power over the individual. We inherited this from Magna Carta and the Glorious Revolution of 1688, and our founders codified it in our constitution. Unfortunately there is no simple word or phrase to name this system of ordered liberty. We often call ourselves a constitutional republic. But what is really important is the contents of our constitution (and how well they are honored), and not merely the fact that we have one. The USSR and NAZI Germany were also constitutional republics.
I can't agree with your assertions. Nazi Germany was a fascist dictatorship - far from the precepts of a republic which fundamentally guarantees individual liberty vs the power of collective institutions.
Rome was a democratic republic for some time, until non-stakeholders were given the same rights as the tax-paying citizens. Greece was a democracy, which equates to "mob rule" or majority rule, vs individual, inalienable rights.
Collectivist societies are diametrically opposed to constitutional republics. Those societies always collapse under the weight of the majority over the minority.