You are correct: Money In Politics is the Underlying Problem.
But on the jury system idea for representation, I have some reservations. Consider the OJ Simpson trial. A jury of ones peers does not guarantee justice.
Another (and also currently unworkable) idea is strict campaign finance reform. Other countries have limited campaigns, strict limits on financing, and limited media proselytizing. Without money driving the political system, those who undertake the difficult task of governing would be more altruistic and show more integrity in their decisions. It is hard to imagine the US adopting this course in the foreseeable future. It may require a devastating failure of the whole political system before it could be rebuilt on a more rational foundation.
Until we take the money out of politics, and end the revolving door of politics/lobbying+”consulting”, we will not have anything close to true participatory democracy.
My advice for the near term is to keep a bag packed, and make sure the passport is still valid.