An Op-Ed in The Times not long ago suggested that the ballot in this country be replaced by "sortition" — appointment by lot, which democratic Athens used in the 5th and 4th centuries B.C.E. to stock its populous Council, Assembly, and jury courts.
While the level of participation was very high, putting to shame our apathetic turnouts, Athenian democracy wasn't at the root all that democratic: Women had no voice, neither did resident aliens, who could not own property; there were slaves, as many as 100,000 in the 4th century, it's estimated in Thomas N. Mitchell's "Athens: A History of the World's First Democracy," and in its Golden Age an aristocratic general and gifted orator, Pericles, essentially called the tune.
Socrates, who was to be sentenced to death for impiety, wasn't a fan, nor were Plato and Aristotle, though he saw some potential good in it. Socrates said, "It is absurd to choose magistrates by lot where no one would dream of drawing lots for a pilot, a mason, a flute player, or any craftsman at all though the shortcomings of such men are far less harmful than those that disorder our government."
It should be added that he thought tyranny and plutocracy were just as bad as mob rule. Nothing could save Athens, he thought, except government by knowledgeable aristocrats trained to govern. The passing of Pericles, who fell victim to the plague, and the Peloponnesian War that he had championed resulted in an oligarchic interregnum before direct democracy was restored in the 4th century, reaching its fullest expression then, according to Mitchell.
The great experiment was laid to rest in 322, Mitchell says, following a disastrous war with the Macedonians, whose chief general at the time, Antipater, required as one of the conditions of peace that only the wealthy exercise political rights. Oligarchy again.
Frankly, I see no reason why sortition would work any better in the United States, a vastly larger country, than the representative democracy (or democratic republic, if you will) that we already have; though it's clear that the Electoral College has skewed things, according to smaller states' disproportionate power, and, because of the winner-take-all allocation of electoral votes, focuses presidential campaigns on battleground states.
The New York Times's Jesse Wegman says in "Let the People Pick the President" that only abolishing the dated system and replacing it with one-man-one-vote presidential elections, as is the case in gubernatorial ones, will assure that the popular-vote winner will win — that the majority, as democracy ordains, will prevail. Hear, hear.