For some reason liberals love to idealize Athens as the home of democracy, our historical tradition and origins. — thedeadidea
Your problem here is that Athens itself did not idealize democracy, so your entire debate is founded on a far deeper falsity than you state.
Socrates, who you quote, was no fan of democracy and an idealist and eventually eschewed. Aristotle is the proper origin of modern democracy (I started a separate thread on the issue of slavery (and AHEM women) on that. Aristotle was a realist, not an idealist, and called democracy the best option among worse alternatives. He was rather more practical minded and cynical and not an idealist at all.
With regard to votes, his debate centers on the benefit of oligarchy versus equality for all, the latter of which is of course, appears controlled by the poor, because there are always more of them. And I will try to summarize it because it is quite long. If you have say 2 rich and 4 poor voting equally, one of the poor quickly realizes they can be more powerful by voting with the rich. If there 2 rich and 6 poor, the poor realize they can split in two and each group can partner either with the rich group or the other poor group to win. And the latter example can collapse into the first, because there 1 person realizes, again, he can switch groups to make a 4:4 split. So what naturally evolves is a system where the rich and poor have equal power decided by a tiny swing group, or even a single person.
Does that, by the way, sound at all familiar to you?
So as to who vote for them, Aristotle agrees, with Plato, that it would be nice if everyone were as intelligent as you, but its impossible. Aristotle disagrees however that a small thinking class should have absolute power, because different crafts and skills earn money in different ways. Plato agrees with that latter point too, but Aristotle's point is, if only a small number of people control the power, they will have a natural bias towards the skills they possess themselves (also a sly stab at sophists with Socrates' view). To assure that the society acts in the interest of all proportionally, all people must have equal control in some aspects of a democratic government.
The issue, Aristotle says, is defining which parts of the system should be controlled more by an oligarchic system, and which by individual equality. So he is more of the opinion that both need to be going on at the same time, which is, in fact what happened in Ancient Greece, and which happens in the USA now too, as its almost entirely based on Aristotle's politics, as far as voting goes (doesnt include the legal wing which was instead defined by Locke's ideas on natural rights).