Your reading of Chapter 8 of The Republic is not very good.
Plato's argument follows by analogy from his tripartite division of the psyche (mind/soul) into eros (desire), thymos (will), and logos (reason). Keeping these parts of the soul in balance would allow one of live with arete (skill).
There are five forms of government put forward by Plato: aristocracy, timocracy, oligarchy, democracy, and tyranny. Each is related to an imbalance of the republic, just as an individual's psyche might be imbalanced.
Timocracy is rule by those with honor (typically military), a trait linked to thymos. Oligarchy isn't rule by the elite per se, but rule by those with wealth (because wealth and honor are conflated in this society). Democracy is a kind of libertine state in which the excesses of eros are indulged. These forms of government emerge out of each other, as timocracy degenerates into oligarchy, which in turn degenerates into democracy, out of which a demagogic tyrant finally emerges to establish tyranny.
Aristocracy, the ideal form of government according to Plato, was one in which the wisest ruled. The wisest, of course, were philosopher-kings—people who had successfully balanced their psyches with logos directing their thymos and eros toward their long-term interests.
It's best not to read The Republic as a direct report on what Athenian democracy was really like. It was a philosophical analogy between the health of the individual's psyche and the health of the body politic.
But it's safe to say that Plato was not pro-democracy. I think one of the more devastating accounts of Plato's political theory is in Karl Popper's The Open Society and Its Enemies.