On the 'random members of the population' thing, I think this could make a very interesting 'second house' (so as far as I understand it, Senate for US and House of Lords for UK). So the detail of bills would still be crafted by a house that knew what it was doing, but then a sort of super-jury would check the bills and challenge things they didn't like. One of the main benefits would be that it would reduce the feeling of 'these crazy decisions, only politicians would make them, no common sense' and ensure that there was a clear sense of what a random bunch of people would do.

With a bit less power/responsibility this isn't quite as vulnerable to the various problems identified: but it could still be very easy to directly or indirectly bribe these people, and you'd still get the '12 men too unimaginative or unimportant to avoid serving on a jury' problem: lots of the most interesting people to have there would have good arguments to not do so