A government. All governments are coercive and premised on a few powerful elites maintaining and exercising their power with a monopoly on violence[0].
Yes, even the "good" ones. At the end of the day government, like war, is a racket.
I don't know that it's the case in any representative democracies - people literally _compete_ to access power.
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.
It is interesting to brainstorm what could be a better system.
I think clear qualifications should be required. So to use the HHS example, it would have criteria to be a doctor, certain years of practice, perhaps practice in several specialties for breadth, some research experience, some administrative experience. You'd want to make the criteria narrow enough that only mostly truly qualified people meet it, but also broad enough that at least tens of thousands of people do qualify for the role, so that someone can't just easily fine-tune their resume to assure themselves a spot. It needs to be impossible to game.
I don't have the answers, but clearly we can do better than the current system where only power-hungry psychopaths choose to travel the road to obtain the vast power they crave.
I'm all for adding a branch that would be primarily picked at random (and I'm all for having a branch that is only recruited among peers of expertise, with fréquent rotation as a way to limit the risk of connivence, etc...)