Athens also had a much different population than we do and not everyone was eligible to be chosen. And I don't think it would go over well today to limit the pool of candidates to exclude those who are nominally unqualified for statecraft.
- You pick 1000 random people to serve as the House of Representatives. - They elect 100 people from among themselves to serve as the Senate. - You randomly divide them into 10 person committees.
If you have enough rounds, there are some great effects:
- The election means that only competent or representative people make it through to the next round. - The sortition makes it so that no one individual or small group is likely to make it all the way to the top.