There are many traps in democracy, but one thing to keep in mind, though it is not always apparent day-to-day is the wisdom of the crowd effect.
I.e. over time, people in aggregate tend to take fairly good decisions.
It might take a while, and there are ample options for suboptimal policies to gel, especially when direct democracy is over-used (Proposition 13 comes to mind).
There is also the risk of tyranny by majority (i.e. ethnic group X in the majority, democratically decides ethnic minority group Y needs special rules). But that's what bill of rights/constitutions/supreme courts are for.
But overall, letting Mr and Ms. Average decide who ought to lead them is far from a bad approach. As per Churchill:
Indeed it has been said that democracy is the worst form of Government except for all those other forms that have been tried from time to time
Arguments opposed to that also brings to mind the fallacy whereas 90% of drivers think they are better than average: "if only people like me got to decide, then we would be in a better place".
A system to determine leadership based on other decision criteria also risks solidifying into a caste/customary system which becomes progressively less suited to changing circumstances, but has a vested interest keeping itself in power. I.e. you may start out with a better system, but you will not stay there.
France for example chose generations of leaders from ENA, a small elite national administration school, whose grads were presumably smarter than average, but it is far from obvious that was beneficial.
p.s. I would absolutely object to having Chomsky in charge of anything in my jurisdiction.