The American system was designed from the outset, on every front, to indirect the preferences of the "people" through multiple layers of middlemen with firmly bounded accountability to those they purport to represent. This is, in the popular consciousness, the salient difference between Athenian democracy and the Roman republic.

It is no longer a popular design, and one that we've been slowly walking back over the history of the union, through general adoption of popular vote to choose electors, direct election of senators, &c.. Nevertheless, the design's legacy persists in all corners of the federal government, so the principle actually does have a lot of explanatory power when trying to make sense of it, even if for no other reason than trying to figure out which parts you want to demolish.

I downvoted it for two reasons:

First, because no polity implements a pure form of any political or economic system. There is no "capitalist economy", "socialist economy", "social democracy", "democracy", "monarchy", etc., just various mixtures of the above. When arguing about what mixture we have, people will often advocate for moving toward (or away) from one of those pure, platonic ideals.

When the US was founded, there was a deep suspicion of unfettered democracy. Political theorists looked for a third way between absolute monarchy and the mob rule of democracy. Athens was a byword for political failure; they were "too" democratic, but Sparta, the Dutch Republic, and the Roman Republic were seen as more useful models, all of whom had strong limits on democracy, and spread power carefully across numerous different competing groups, many of which were chosen via methods other than direct democracy (eg, heredity, lifetime appointments, geographic distribution, wealth, etc.).

So when people say "the US was founded as a republic, not a democracy", what they're referencing is that old debate. They're saying "remember, we're meant to be looking to Sparta not Athens as our model", which is true. The fact that Athens and Sparta also had many similarities, and that you could also claim (correctly) that Athens and Sparta were examples of the same political system is irrelevant, because they're still not identical. The distinction has meaning.

Second, because you said:

> What those that push the slogan dont tell you is that they don't want a republic either, what they really want is a theocracy or some kind of corporate dictatorship.

That is not true. Hell, you can agree (as I do) with the observation that the founders carefully worked many anti-democratic ideas and systems into the US constitution (no direct election of senators, no direct election of the president, strong weighting of political power to rural areas, etc.), and then also think that this no longer makes sense, and we should continue undoing them. A perfectly valid response to "the founders wanted us to be like Sparta not Athens!" is "true, but Sparta was a horrifyingly violent slave state that ruled via terror, then collapsed into irrelevancy when they could not adapt to new technologies and geopolitical forces; the founders were wrong".

So no, not only does everyone who push that slogan not want to install a theocracy(!), some merely want to roll back a few recent reforms, others want to maintain the status quo, and some actually want to continue the push towards greater direct democracy, but just have no interest in pretending there's if you use a secret decoder ring it turns out Thomas Jefferson and Hamilton and Adams secretly agreed with the exact policies that progressive reformers are pushing in 2021.

I downvoted it for mind reading: "...what they really want is a theocracy or some kind of corporate dictatorship".

The two terms have obvious signaling implications because the two major US political parties are called Republicans and Democrats. The popularity of the statement is just as a slogan for Republicans who like the way the implication sounds. It's the same as the Democrats who complain that the electoral college is "undemocratic" despite the conspicuous amount of voting involved. It's just banal sloganeering.