Technically, "idiot" (or at least the Latin "Idiota") just means a layman, someone without a government job. I don't see a problem with the population being mostly private citizens.
Democracies are usually run by ignorant masses, which is why the US was set up not to be a Democracy. Democracy in the mind of the founders equaled "mob rule." The Republic was set up prevent that. As the founders predicted, faction essentially has lead to the death of the republic, albeit more subtly than they expected. The system lumbers on, but has never served the public good particularly well.
I agree with Madison that you cannot eliminate faction while preserving freedom, but there is a lot you can do discourage it. First, forbid the government from subsidizing any parties or primary elections. Second, set up a voting system that discourages primaries and gives individuals a greater say in how they vote, rather than needed to pledge their full support to one leader. I prefer allowing fractional ballots, but I'd be fine with Range Voting too. Either method would make voters have to study more candidates and their positions, and make polarization much harder. Making voting more complicated and eliminating cognitive shortcuts like party identification would also discourage stupid people from voting, and allow the better educated to better express what they want. It would also make it harder for candidates with extreme views to get elected.
I'd definitely also want to allow for national initiatives (probably requiring significant super majorities), including the ability to overturn any law, executive order, or judicial doctrine, and to remove any government official from office.
Ideally, I'd also like to significantly restrict citizenship. No one should ever gain citizenship automatically, whether based on where or to whom he/she may be born. Emigration would not be restricted at all for non-criminals, and any non-citizens could be deported/banished for crimes. Everyone would have to be naturalized, which would require passing tests in History, Logic, Mathematics (at least basic algebra, preferably basic Calculus), Economics, Ethics, English (I'd much rather require Latin instead, but for the US using English as an official language makes more sense), and Political Science. (There may be a small number of exceptions of those who can become citizens wihout such tests, but these should be pretty much limited to war heroes.) (Public Education in these areas would be available to anyone for free, regardless of age, but be non-compulsory. I'd also have easily accessible, government-sponsored news media providing info on all the candidates and initiatives, but not try to stop private media from doing whatever they want.) Government could then function as a more valid social contract, to which citizens knowingly submitted themselves.
