Some of the examples I've seen it tried - I've seen the people setting it up trying to fix the outcome by carefully choosing the question, then providing expert advice on options scoped by the question.
Framing of the question is a powerful tool to promote the outcome you want, and avoiding ever asking certain questions is another.
Not saying it doesn't have it's place - you just need to be careful that the process isn't used to try and legitimise what would otherwise be unpopular policies via concentrated persuasion on a small number of people.
Framing questions is already a problem with legislation. You can frame "do you want to increase online surveillance" as "do you want to protect children" very successfully!
If you are saying choose people at random to be an MP for 5 years ( or whatever ), then sure that's different and it would be an interesting experiment - though that would be a pretty stressful job to pitch people into at random.
It would be interesting to see how those random 600 people would organise to get stuff done. In the current government you have specialisation - home secretary, foreign secretary etc - you wouldn't want to keep that structure and randomly allocate roles - but if you have the 600 vote on everything then you still have a bandwidth problem.
It may work in some other country..
Jury service in the UK is generally seen in a positive light ( despite having far too much hanging around ).
I suspect the US problems could be easily fixed by forcing employers to pay you while you are doing it.