This goes beyond even how we usually talk about the distinction between Direct Democracy and Representative Democracy.
Even the most direct Democracy still needs magistrates of some sort to enforce laws and manage things, and make quick decisions in a crisis when there isn't time to debate. In classical Athenian direct democracy these magistrates were not chosen by voting but by the Lot (Sortition), meaning a qualified citizen was selected at random.
The never implemented hypothetical constitution proposed by Hippodamus of Miletus is described as "less democratic then Athens only in that magistrates were chosen by Election rather then by lot". In other words it's not just that Elections weren't required for Democracy, they were contrary to it. Sparta is thought of as the antithesis of Athens and in turn the most anti-democratic City-State of Classical Greece, but Sparta did have elected magistrates in the Ephors, and they were often the actual Power in Sparta rather then the Kings.
Athenian Democracy considered Elections a threat because they didn't want one person to become too Popular, Popular enough to form a popular Tyranny, and choosing magistrates by popularity inevitably leads to cults of personality.
The evangelists of Athenian Style Democracy considered the accountability they hold magistrates to vital. Elections make accountability more difficult, when everyone in power is also basically the face of a segment of the population holding them accountable for even the most basic of wrongdoings becomes politicized. We see this in modern US Politics, in 2016 it was considered dangerous for Trump to make even the suggestion of criminally prosecuting Clinton a campaign promise, but now post January 6th a lot of the same people think it's a dangerous precedent not to prosecute Trump even as he's officially running for president again.
When people accuse modern western Democracies of actually being Oligarchies they usually mean that in the sense of how the Rich use their Money and influence to undermine how the system is "supposed" to work. However Oligarchy as a Greek word is not inherently synonymous with rule by the Wealthy (that would be Plutocracy), it simply means rule by a small group. Meaning even if American Democracy did work exactly how it claims it's supposed to, that would still be Oligarchy, that would still be rule by a small group rather then the masses, membership in that small group being decided by winning popularity contests (or being appointed by a winner of a popularity contest) doesn't matter, especially when there is no real accountability for representatives who brazenly defy the will of who they represent. Whatever legitimacy Representative Democracy used to have was destroyed in the Anglosphere by Edmund Burke.
And as long as there are some people who are massively more wealthy then most people, they'll find a way to influence and control the representatives. Any "campaign finance reform" you pass to address how they're doing it now will only result in them changing their methods.
Now one difference between Classical Athenian Democracy and ours definitely doesn't make theirs look better, and that is the restrictions on who could vote, women, slaves and non native residents were all excluded. However the United States started with all those same restrictions and more, the Athenian Constitution to which I refer never had property requirements, but the U.S. originally did. We also started with a from of Slavery far more brutal and dehumanizing even then how Sparta treated the Helots much less Athens where most Slaves were just unpaid Butlers and Maids. Women didn't get to vote for over a hundred years, and we still haven't enfranchised all non native born residents. Plus we take the Vote away from convicted criminals permanently even after they've paid their debt to society.
There is evidence that there were people in Ancient Athens who sought similar reforms, perhaps the people being satirized by Aristophanes in The Assemblywomen, if the Athenian experiment hadn't been cut short by being conquered by Sparta, Macedon and Rome perhaps those reformers too could have succeeded in time. And then in the Hellenistic Era we get the Stoics who I've already talked about on this blog.
