I imagine the random person, if they were to accept, would likely find themselves out of depth and simply outsource most of the job to whoever convinces them they're the right choice.
The system would then morph into a variant of representative democracy. Instead of millions voting for, say, around a thousand of various level representatives, you get a random thousand "voting" for "consultants".
These consultants would be directly hired by the random thousand to do the work for them. They would predicably have marketing campaigns to make themselves seem the right pick. They might even offer their services for free. Demagoguery and corporate sponsorship as usual.
In the end 1000 people would hire, say, 1-1000 consultants to represent them. Thus emulating the original process, or worse. The only difference is the tiny electorate, which increases volatility.
If you'd like to regulate it to prevent this, you'd have to expend an effort and fight similar pressures as with doing it in any other system.
The system would then morph into a variant of representative democracy. Instead of millions voting for, say, around a thousand of various level representatives, you get a random thousand "voting" for "consultants".
These consultants would be directly hired by the random thousand to do the work for them. They would predicably have marketing campaigns to make themselves seem the right pick. They might even offer their services for free. Demagoguery and corporate sponsorship as usual.
In the end 1000 people would hire, say, 1-1000 consultants to represent them. Thus emulating the original process, or worse. The only difference is the tiny electorate, which increases volatility.
If you'd like to regulate it to prevent this, you'd have to expend an effort and fight similar pressures as with doing it in any other system.