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 ineffectual or one being used/manipulated.

I could also envision an endless cycle of elections with 75%+ of the population voting "none of the above" because of issues like "Not my personal favorite candidate" or "eats the wrong mustard" or "I hate the idea of government"