It would save a lot of computing power, but it doesn't solve the core problems of defeating double spending and attracting more compute power to the network to check against that spending. Even if there were random elements, I think peer-to-peer voting(even if done by the software itself) would be too easy to game and unduly favor dishonesty and cheating. It would make trust = wealth, which is a fallacy because what if all those private keys contained stolen bitcoins, etc?