Imagine if you will that every voter had the ability to submit their idea's for other voters to support/debate and the top ones go onto being voted into policy. For some they could just vote upon the policey, if they choose, be their choise.
But that would take technology that was trusted. But then voting democratically is something that happens in so many countries, be great if there was an open/auditable standard of such a system that the people could get behind. Facebook and other social media is not that solution.
Today we have a token say periodically with those representatives being driven by media, that is in part driven equally by events and social media reaction to those events. It is messy, informal, rabble driven mess and that is what plays out in politics. Yes that needs to change, but it needs to change slowly to allow people to adjust or you will create just as much friction as you are trying to solve.
But for letting goverment being decided as a lottery, well - look at jury duty, that is how that works. You can find good examples of this as you can bad examples. So on average you could say that you would be no worse off. But the extreme's could potentially be larger. Let's face it, what are the odds of a lottery picking all KKK members as a government - they are not zero. That's all that needs to be said upon that idea.