I like the idea, but I worry about choosing random members of the public when so many people are unprepared for it. Any kind of government made up of "the people" requires that those people be literate, educated, and informed. With things the way they are today I'd worry that your secondary elected legislative body would end up doing everything and you'd either end up with a figurehead who'd be out of their depth and either ineffectual or used/manipulated, or worse and endless cycle of elections all ending with a 75%+ of the population voting "none of the above" because of issues like "Not who I personally would rather have had" or "eats the wrong mustard" or "I hate the idea of government"

Nice one, that might actually work. But it will be hard to explain to the electorate.