"Democracy" just means "majority rule." In a democracy, the majority can trample the rights of the minorities. In a constitutional republic, those rights are protected by a constitution that is very difficult to amend.
America today is somewhere between a true democracy and a true constitutional republic. Officially, it is a constitutional republic, and most of the key individual rights are protected (free speech, etc.). Unfortunately, there is no separation of state and economy, and your wealth can (and is) seized arbitrarily.
There is also a very large spate of regulations that severely curtail your proper right to act as long as you don't violate the rights of others.
So basically, the mechanism of a constitutional republic still exists in America, but its actual purpose has been sundered.
This split between "democracy" and "constitutional republic" is something being pushed more and more, but it's really a fantasy peddled by some US-based politicos with vested interests in devaluing the concept of democracy and making you accept the fact that you should be happy without it.
A "constitutional republic" is something defined by a set of laws with certain specific elements in common: having a constitution and being a republic (which is also quite a loose term, used to define "anything that is not a monarchy" -- Iran is also a constitutional republic, for example, albeit a theocratic one).
A "democracy" is not a comparable entity. Can you please point out any country who defines itself, in official terms, as a "Democracy"? Even classic Athens didn't call itself such a thing. This is because democracy is not a set of laws: it's a declaration of principles, an abstract utopia of where your laws should come from (i.e. "the people"), traditionally associated with some sort of voting system. On the formal level, there is no clear definition of a complete form of government called a "democracy"; it's commonly assumed such a thing involves some sort of voting rights (better if universal), but that's about it. The term is used to indicate the utopic state where government is carried out "by the people" in its totality. There has been no such perfect government throughout history and probably there never will be, but it's an utopia from which real forms of government can find justification and inspiration.
You can compare a "constitutional republic" to a "constitutional monarchy" or an "absolute monarchy": they are all specific forms of government with very specific common formal traits. A "democracy" is a completely different concept. You live in a constitutional republic based on democratic principles, i.e. a democracy... albeit a flawed one, like all of them are. You should strive to make your republic as genuinely democratic as possible, not renounce your rights "because after all we've never been a democracy".
I know more history than the average person, and I totally, utterly disagree with you.
From what I understand, the US Founding Fathers explicitly wanted to avoid a democracy, which was seen as an unjust system that punished Socrates and was associated with the downfall of Rome (at one point, the Roman emperors apparently had to essentially feed the populace in order to maintain enough popularity to stay emperor, bankrupting the state).
Rather, they wanted a system that protected individual rights. You know, life, liberty, and the pursuit of happiness.
That is explicitly opposed to a system of rule by the majority or rule by the people.
The US was not supposed to be a democracy.
And regardless of the historical details, why would I want to be ruled by the majority? I want to rule myself, and I have no desire to rule others. That's the only morally proper attitude.
Regardless of what they were trying to do, the founding fathers created a democratic system -- who is "we the people", if not literally the "demos" ?
Are laws not written in an assembly by majority vote?
Are you not electing representatives by majority vote?
And this has always been the case in the US system.
See my other post here about the different types of majority and how they can represent different things and/or failing at representing the will of the demos. The fact that the founding fathers were trying to avoid the worst elements of democracy doesn't mean they didn't want a democracy at all; if that was the case, they'd have just nominated a new King and be done with it. No, they were establishing a government by "we the people", which is the exact definition of a democracy. Whether or not they misused the term to indicate the "tyranny of majority", which is a separate concept, does not mean that we should keep misusing it in modern scientific and political debate.