This semantic argument is completely tiresome. There's no contradiction between being a democracy and being a republic. The US is a democratic republic. Non-democratic republics include ancient Rome and 18th-century America (where slaves couldn't vote). Democratic non-republics include the UK, Canada, and Holland (which are monarchies).
"Republic" doesn't actually mean anything aside from "not a monarchy". When the founders of the United States wrote movingly about being a republic, they were firstly trying to emulate Rome and secondly trying to emphasize the then-popular Enlightenment ideal that monarchies were outdated and it was better to govern explicitly in the name of the people.
(The word "republic" emulates Rome because it's the term used by and for the Roman state. It's derived from the Latin "res publica", for "public property", which is exactly synonymous with the English word "commonwealth".)
I'll also add that democracy does not imply elections.
Actually, in Athenian democracy, decision makers were elected through sortition [1] (drawing of lots).
They considered elections to be the very mark of an oligarchic system, which is what (in my opinion) most so-called "modern democracies" have become. (As illustrated by the importance of parties and of a class of professional decision makers, namely politicians, who have incentives to increase their own power and limit the number of people who reach their status. Usually this is done by rigging the electoral system.)
To be fair, pure democracy in Athens led to all kind of disasters as the crowd was easily swayed by demagogues like Alcibiades, leading to disastrous foreign adventures like the sicillian expedition in the Peloponnesian war (A war like no other is a great book on this). It's hardly a model we should emulate.
I prefer the mediated kind of democracy (in spite of the tendency to oligarchy) as at least it tempers the rule of the mob.
There is no contradiction between being a democracy and being a republic however, as you add "constitutional" in front of "republic" it becomes a constitutional republic, which is not a democracy.
Can you elaborate? It's pretty simple in my opinion: democracy is the power of people and in a democracy (which would be also a republic, since it's not a monarchy or a theocracy) the people is the ultimate source of power. That could be either a direct democracy (where all people get to decide) or representative democracy, where the people choose representatives to decide everything. There is no power above the people's in a democracy.
In the USA the ultimate source of power is the Constitution of the US. It rules over the power of the people. Since the ultimate source of power in the USA is not the people it cannot possibly be a democracy.
I will skip the nonsense point of me being somehow stuck, if you think more than 5 seconds I believe you can figure out how this works yourself.
As for your second point: it contradicts reality. There had been even direct democracies (ancient Athens among many others) and, judging by the frequency that constitutions change around the world in most democratic countries you need just a referendum to entirely repeal the constitution.
Since the American democratic process is capable of passing arbitrary Constitutional amendments into law, and since the Constitution itself says its establishment derives from the will of the people, your interpretation is completely untenable.
Of all the hundreds of democracies established throughout history, almost all of them had governance through elected representatives and almost all of them had some kind of legal limits on what could be done through the democratic process (at least not without using some other democratic process to revise those limits). At that point, if you don't want to count them as real democracies, feel free to continue inventing your own language because you're not speaking the same English as the rest of us.
I'm sorry, I just don't see where you're coming from. By the most common usage of the word, most countries deemed "democratic" have some type of constitution, and many of them even have things like monarchies that are nearly impossible to abolish according to the written law--through there's a practical understanding that if the people wanted to, they could do away with these things.
On what authority was the Constitution itself established if not the will of the people? On what authority does the Constitution assert its own if not, in its own words, "we the people"?
I contend that the people of the United States could do away with the Constitution the same way they did away with the Articles of Confederation. The Constitution even gives them a mechanism: pass a Constitutional amendment that repeals every preceding amendment and article and establishes a new Constitution on top of it. Once this legal maneuver was done, no one would ever need to read the original Constitution and its authority would be moot.
In fact, if you consider the idea that the Articles of Confederation were the original constitution, the fact that they are no longer in effect today even absent a formal mechanism in the Articles to abolish them shows that the Constitution could be replaced in the same way. "We the People" somehow had the authority to establish the Constitution in 1787--on what grounds do we not have the authority to establish a new one should we see fit?
It was a Constitutional Republic.