> I haven't got the first clue about governing a country

Is this really so different from quite a number of high profile politicians today? Many are mostly good at networking and how to use the media machine. The actual competence is with the invisible people behind them, and the bureaucrats. I see little or no difference, even disregarding current administrations (not just in the US).

…yet we trust juries.

Most people are idiots and trust things they shouldn't. Lee Kuan Yew doesn't trust juries.

Same as what we have now but I think the moral filter applied would be stronger coming from a random citizen as opposed to a power-hungry citizen. Also, with randomness over the whole population, you can be sure everyone in congress and senate would be different every election cycle. So politicians won't be able to stay in office long enough to form meaningful alliances; they will not trust each other and thus watch each other better to prevent corruption.