You seem to be of the opinion that anyone can competently lead. I do not believe this to be the case. Imagine your lottery picks some jobless guy with mental issues. How about racists or extremists that will make next four years hell for some people they don't like? How about selfish "screw everyone as long as I get rich" types?
OK, we need to now sort the people into electable/not electable. The criteria would already be a thorny issue. How do you judge competence for such a position? If you seek experience of leadership, this will effectively swing the balance to already rich and powerful.
Even if you get a good, intelligent, charismatic, altruistic leader: After four years the next one most likely will undo or heavily modify the policies set. After all each leader wants to leave a mark, especially if he/she cares about some specific issue. No controversial, but necessary public project will last longer than four years!
This four years thing is already an issue in this day: Why should the leader ever care about long term if he will be long gone by then.
I am sure there are other problems, but I do not have any answers or ideas. Social structure, fairness (both short and long term) and ethics are so complex. Especially since every single actor in the system is inherently selfish and greedy.