Mathematicians already know that random selection is a powerful tool in fairness in electoral processes, the problem is with asking the modern democratic citizen and politician to accept a random process.
Also ensuring the legitimacy of the random process (which is not really about technical features, but about social consensus): just how much easier is to claim a PRNG (or a rabbit that chooses a box, or paper slips inside cookies) is rigged.
Yep! For a modern day example, there's a famous conspiracy [1] the Champions League draw, which involves pulling balls from a bin, that certain balls are warmed or cooled so that the ball-chooser knows to draw them in a certain round.
They could rid themselves of this conspiracy by randomly selecting the ball-drawer (instead of using a paid random ex-footballer), but they don't. :)
