You're right, that was a bad example.
So let's look at Athenian-style direct democracy. You have the ecclesia, an assembly where any male citizen can vote on proposals and elections (like a more ad-hoc city council meeting and election in one). You have a bureaucracy of 500 randomly selected men who run the city government. Once a year, you can vote to exile any leader for 10 years. Slavery was big business and women had no rights. There were very few elected officials and they were chosen randomly for the highest offices. Think of random people with conflicting, crazy ideas being voted in and out regularly, as judges, as bureaucrats, as heads of state. There was uneven representation because you had to show up regularly to voice your concern, only about a quarter of the people had this time/convenience luxury, and of course most people weren't very educated, so their opinions and positions were often.... dumb. But anyone who was a male citizen could show up and vote on just about anything, and simple majority was the winnter. The assemblies weren't held accountable to themselves, and could vote to break their own laws. Trials lasted one day; arguments were given, and then an immediate vote by the jurors. Out of 1100 citizens chosen by lot to (effectively) govern, 100 were elected. There were age restrictions and short term limits.
It changed over time. Over 300 years they went from having a king to having 3 Presidents who were aristocrats elected every 10 years, to elected every 1 year, to literally assigning the Presidency to a random citizen by lot, with the military overseen by 10 elected generals with varying degrees of social/political power in addition to total military command. If the people didn't like the job they were doing, they'd be voted out, fined, possibly executed. Originally the 500 civil servants were chosen by class, with the highest class (richest) getting the top positions, and the poorest class couldn't be selected; but later they could be selected and even got paid (partly because the "random selection" just happened to represent a lot of rich dudes). Eventually laws were not just things decided by assemblies, but small pools of lot-assigned citizens who decided what the law would be.
Some researchers have suggested that direct democracy like this wouldn't work over a very large area. It can work in towns and cities, and small city-states, but there's just too many factors for it to be practical in a large scale diverse modern nation.