First, I think comparing our concept of democracies with athenian democracy is rarely a good idea. They didn't go for elected officials in the way we do - even our most extreme concept of direct democracy would be, for the athenians, an oligarchy, because the rich and famous could use these traits to secure general approval and political power.

Second, it's worth noting that most records of Athenian democracy come from its detractors. People like Thucydides were basically members of an oligarchic club who consistently conspired with the Spartans against Athens.

I don't know if direct democracy is unstable. The most direct-democratic nation in Europe is Switzerland, and they are so stable they're basically stationary. I don't even know if Athens was particularly unstable for its time-period - it's not like the surrounding city-states, Sparta included, had a particularly good decision-making record.