Given how much effort people put into getting out of jury duty I can only imagine how something like sortition (which I think is what you're describing) would work. Would it be like military conscription, where you could go to prison if you're AWOL? If so, who really has the power in that scenario?
And there's also the problem of competence. Forcing some random to act as Secretary of HHS is more likely to get us an RFK type than someone who knows what they're doing. But then vetting candidates with some kind of voir dire like system is just voting, except only a few people get to do it.