The whole “we are a republic, not a democracy” argument is premised on a silly assumption about language and a purposeful misreading of definitions.

James Madison (in Federalist Number 10 that you quote) made clear how he was defining a democracy vs. a republic:

“The two great points of difference between a democracy and a republic are: first, the delegation of the government, in the latter, to a small number of citizens elected by the rest; secondly, the greater number of citizens, and greater sphere of country, over which the latter may be extended.”

So if your point is that we don’t have a “democracy” in the style of the Ancient Greeks where everything was done by national plebiscite then you are 100% correct.

This is where the lazy language comes in. No one uses the term “democracy” as James Madison defined it in Federalist #10. In common usage, democracy does not mean “the citizenry votes directly on all issues.” Rather it means that, generally, the citizens vote and choose their leaders. So if you believe that language is defined as how words are actually used, the “we are not a democracy” argument falls apart.

Even if you are a formalist who believes that that the meaning of a word is defined by the dictionary and not common use, you are wrong. Look up a dictionary definition of democracy. It is not Madison’s definition and it is hard to see how the United States doesn’t fit.

Lastly, what the heck do you even mean that we are a Republic not a democracy? What does that tell us about anything?