I think it makes sense in terms of a multicameral legislature. I can't imagine you want to pick the president by lottery or probably not even senators but why not the House of Representatives? Maybe we need a lawyer-heavy legislative body to write laws, but we also need legislators who aren't lawyers so people believe somebody in the legislature has some empathy for them.
I think that it is for the president where this would be most beneficial.
For senators and representatives there still is a little randomness, but when I look at many presidents that have been elected in recent years in various countries, it is more than obvious that any randomly chosen individual would have been almost certainly a less bad human and more suitable for the presidential position.
After a president is chosen by lottery, there should be a rejection procedure, to eliminate any really inappropriate candidate, but then a new candidate should be chosen by lottery to replace the rejected one, until no reasons for rejection can be found.