Sabine Hossenfelder has a video, Is the USA a Democracy or Republic?. I confess I'm inclined that this is just not a discussion with which people who are not American citizens should bother themselves much at all. As people should know by this point, the US has a distinctive political culture that, despite its immense influence, has its relatively unique quirks, and one of its quirks is that we often use political terms in unusual ways. For instance, we still often use terms in senses that are a couple of centuries old and not necessarily informed by the ephemeral political happenings of five minutes ago. even when people in other political cultures have passed on to other things. The use of 'democracy' in the statement "The USA is a Republic, not a Democracy" is sometimes, depending on the context, a good example.
Taken in this deliberately old-fashioned sense, the statement is an old idea in American civic history. The opposite is also an old idea; you find people arguing about it, in broadly similar terms to those in which they argue about it today, in the first half of the nineteenth century. The reason for the matter is that 'democracy' is used for very different things. Hossenfelder takes it to mean representative government involving elections; this is what in the United States would traditionally be called a republic. That point goes back to the roots of the country. We find this sort of discussion in Madison's Federalist 10, for instance:
From this view of the subject, it may be concluded that a pure democracy, by which I mean a society, consisting of a small number of citizens, who assemble and administer the government in person, can admit of no cure for the mischiefs of faction. A common passion or interest will, in almost every case, be felt by a majority of the whole; a communication and concert results from the form of government itself; and there is nothing to check the inducements to sacrifice the weaker party, or an obnoxious individual. Hence it is, that such democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security, or the rights of property; and have in general been as short in their lives, as they have been violent in their deaths. Theoretic politicians, who have patronized this species of government, have erroneously supposed, that by reducing mankind to a perfect equality in their political rights, they would, at the same time, be perfectly equalized, and assimilated in their possessions, their opinions, and their passions.
A republic, by which I mean a government in which the scheme of representation takes place, opens a different prospect, and promises the cure for which we are seeking. Let us examine the points in which it varies from pure democracy, and we shall comprehend both the nature of the cure, and the efficacy which it must derive from the union.
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.
The republic vs. democracy notion here is a particular adaptation of an older view, ultimately derived from the classical period, when (for instance) elections were taken to be obviously anti-democratic by nature, since election, unlike sortition, creates oligarchies of people (usually rich and connected pople) capable of being elected. (The second of the 'points of difference' would have been extremely controversial at the time, but has arguably not been controversial in the context of American civics since.) Madison takes it to be quite important that we are a republic rather than a democracy.
That we are quite so was famously put into question by Alexis de Tocqueville in Democracy in America (Book II, Chapter 1):
The American institutions are democratic, not only in their principle but in all their consequences; and the people elects its representatives directly, and for the most part annually, in order to ensure their dependence. The people is therefore the real directing power; and although the form of government is representative, it is evident that the opinions, the prejudices, the interests, and even the passions of the community are hindered by no durable obstacles from exercising a perpetual influence on society. In the United States the majority governs in the name of the people, as is the case in all the countries in which the people is supreme.
But of course, Tocqueville takes it to be true that the 'scheme of representation' makes a country a republic; what he is arguing is that despite having the form of a republic, for practical purposes, the United States is a democratic wolf in republican sheep's clothing, subverting republican forms by democratic practices.
Likewise, the United States Constitution never mentions democracy, but it does explicitly require Congress to guarantee to each state a "Republican Form of Government" (Article IV, Section IV), which has always been interpreted as a government like that presented in the Constitution itself. Of course, you might see 'democracy' in the 'We the People' of the Preamble, which meant the people of the United States organized as states. That requires a conception of 'democracy' in which it means a representative scheme of government, as I talk about below.
None of this is to say that we don't use the term in somewhat newer senses, although even then Americans tend to like old-fashioned usages. Some people have sometimes characterized the United States as a democratic republic; this classification is to take into account that we are more like Switzerland than like the Republic of Venice or the Republic of Rome, which are then characterized as aristocratic republics. But this, too, has at times been a controversial designation. While we may be more like Switzerland in some ways, we are also in some ways more like Rome than like Switzerland, and it is not completely unheard of in American history for people to argue that we are in fact an aristocratic republic with significant democratic elements. That would perhaps not be surprising, because we were deliberately designed to be a mixed government, and some people have argued that we have a very mixed-up government.
You also find views that claim that we are a constitutional republic rather than a democracy; the idea being here either (depending on the case) that the United States as a people is formed entirely by the U. S. Constitution and therefore has no authority beyond it, or that while we have a popular government, the power of the people as such is exercised entirely in giving legal effect to the U.S. Constitution. Such people want to say that if anything is attributed to the people of the United States, you should ask where in the Constitution it can be found. In these senses, calling the U.S. a 'democracy' is often seen as an attempt to subvert the Constitution of the United States by an extralegal authority that is called 'popular will' or some such in order to make it sound impressive. Historically Orestes Brownson might be an example of someone rejecting democracy in this sense, but you do find people occasionally using the terms in something like this sense even today. And it is a matter that can be genuinely of concern; one regularly finds 'republics', really dictatorships, where 'the will of the people' is used to ignore the legal rights of actual people.
Franklin D. Roosevelt pushed a conception of 'democracy' where it was represented in the United States by the Bill of Rights (which mentions 'the people' and their rights and powers five times). This was part of his rhetorical push to stir the United States against the Nazis and the Fascists, and this is a common use of the term even today. In this sense it is not a specific form of government but simply the recognition of rights of the people.
If, of course, you want to claim that 'democracy' means representative government in which each eligible citizen gets a vote that is counted as long as it complies with the relevant election laws, then of course we are in many ways a democracy. In this sense, the Office of the President, the Senate, the House of Representatives, the Electoral College, and significant parts of our state and local governments are democratic simply by their nature. You will quite clearly find people claiming that the Presidency, the Senate, or the Electoral College are not democratic; they quite clearly do not mean by the word, "representative government in which each eligible citizen gets a vote that is counted as long as it complies with the relevant election laws". What they do mean is very hard to determine. Sometimes they will appeal to what is sometimes called 'voting weight'; I've noted before that there is no single universal measure of voting weight, so what they mean by that has to be guessed from context. Likewise, sometimes they talk about voting weight in terms of the (generally undefined) slogan, "one person, one vote", but this is extremely misleading -- all of our election systems are 'one person, one vote' in the sense that it's illegal for someone to vote twice and are not 'one person, one vote' in the sense of every person being able to vote, since voting is confined to legally eligible citizens who have registered in a voting district. The slogan is really a veiled attempt to say that votes are not being treated equally in some particular sense that the person using the slogan thinks is important.
Sometimes by 'democracy' people have meant what has more recently come to be called popular majoritarianism. In this sense the United States is definitely not a democracy; very large portions of our government, at both the federal and the state level, were designed specfically to oppose popular majoritarianism, and are what people sometimes call electoral majoritarian, which, indeed, some people call 'republican'. Sometimes when people say, "The United States is a Republic, not a Democracy", this is precisely what they mean.
All of these have their legitimate place. But frankly, I think it's also true that many times when people talk about 'democracy', they have no actual idea what they mean; the conclusions they draw from their claims about democracy are often incoherent. It is a placeholder term and they attribute a number of things to it without worrying about whether the term can bear all the attributions consistently. This seems less commonly the case for 'republic', but I would not be particularly surprised to discover similar incoherent uses on that side, either.
Thus when people say things like, "The United States is a democracy", or "The United States is a republic, not a democracy", you should usually ask what they mean, not assuming that you know already. Or, I suppose, you can just nod politely and move on to talk about other things, depending on the context.
