A very important point in politics that doesn't often get discussed is that simply holding a legal office doesn't give you any actual powers. The most likely result of a sortitionist system would be that the bureaucracy controls the state, and that the officially appointed leaders are strictly figureheads.

Despite what the law says, people in the administration don't actually have to follow the orders of their appointed superior. If they feel that the senator/congressman/president/etc can be fooled, or simply that enough of their colleagues are on their side, they can just ignore even direct orders. What's the senator going to do? Call the police because their subordinates are ignoring them?

So one of the most necessary qualities in an official is being able to control the levers of power - through connections, charisma, physical power in ages past, money - whatever it is, you need your subordinates to actually do what you tell them. Government isn't a boardgames where people follow the law just because those are the rules of the game.