Is random selection of leaders really an effective substitute for voting? Are elected leaders, in the aggregate, consistently worse than 50/50 chance? Random selection must be even more contrary to aristocracy than voting, because voting permits the vestiges of aristocracy to persist. If leadership selection were really random, you'd expect aristocrats to be elected equal to their percentage of the population. It wouldn't entirely disenfranchise them, which is to say they wouldn't be treated the same way they've sought for generations to treat the majority.
But then, do we really understand randomness? It's a really interesting idea, but I sooner expect aristocratic tendencies to blow up the world before allowing any further diminishing of their power.