Your usage of "republic" earlier seemed to be premised on an argument centering around the design and function of Federalism in American governance, which in turn implies certain arguments regarding how American political culture leverages political structure to address potential (and manifest) pathologies in simple systems of majoritarian rule.

Otherwise, your use of "republic" makes little sense in the context of this thread as the principal and general meaning of the term is a contradistinction to monarchy and similar systems--e.g. Roman Empire. Early American writing tended to conflate "republic" and "democracy" because in the context of contemporary 18th century European political discourse they were as a practical matter two different ways to describe the same systems; there was little need to make finer distinctions. Today we understand nations like the U.K. or Finland as both democracies and [constitutional] monarchies, but that shift occurred later. Similarly, as more identified modern democracies came into being, they created contrast to the particular structures of American government.

Importantly, "republic" and "democracy" were not mutually exclusive or in tension. The John Birch Society created the modern meme of "the U.S. is a republic, not a democracy" as a way to embody and emphasize a particular set of political arguments regarding the proper way to understand the design and function of American political institutions; that proper way becoming a premise to arguments opposing reforms which were popularly deemed more "democratic" in the contemporary narrative. (IOW, rather than trying to reappropriate the meaning of "democratic" from liberals, they decided to run in the other direction.) Those arguments were motivated by their notorious opposition to the Civil Right movement and [claimed] racist and sexist ideologies.

But I didn't mean to suggest that you or others share the latter ideologies, or even knew about their historical association to the meme. The meme has succeeded in part because it aligned well with broader, subsequent evolutions in American conservative, especially libertarian, thinking. And it survives as a pithy way to oppose certain "democratic" reforms by alluding to a general set of arguments emphasizing the role and importance of political structure, and in particular the specific arrangement and form of American political institutions. Though in its modern form it arguably only begs the question--which aspects of structure matter, and why, precisely, should they be privileged in the particular situation under discussion. Conversely, claims of a reform being more "democratic" beg similar questions.

For people who have yet to encounter the meme, the usage can be confusing, presuming they're even able to identify the conflicting meanings of "republic" at play. Today these people are often directed to web sites or videos that "teach" the meme as an historic fact of American political philosophy, often paraphrasing or directly quoting original John Birch Society literature and media--though almost never saying so, perhaps because the author may not have even known the provenance of their source material.

To be clear, there's no denying that political structure is and always was heavily emphasized in American politics. Federalism is embedded deeply in the American Constitution, as are other organizational structures at both the state and federal level. But "republic, not democracy" is a modern, manufactured dichotomy designed to package and sell a particular set of political arguments, albeit arguments which have become more general and amorphous over time.