Random selection is necessary to create stable meritocratic institutions

There is an interesting example of random selection of leadership from the Bible when the apostles replaced Judas. The criteria were agreed upon and then lots drawn.

Acts 1:21-26 Therefore it is necessary to choose one of the men who have been with us the whole time the Lord Jesus was living among us, beginning from John’s baptism to the time when Jesus was taken up from us. For one of these must become a witness with us of his resurrection.” So they nominated two men: Joseph called Barsabbas (also known as Justus) and Matthias. Then they prayed, “Lord, you know everyone’s heart. Show us which of these two you have chosen to take over this apostolic ministry, which Judas left to go where he belongs.” Then they cast lots, and the lot fell to Matthias; so he was added to the eleven apostles.

Can you imagine this practice replacing the Papal conclave? Or, pastor selection at your favorite Protestant group?

seems quite meritocratic with a pinch of (Lord's) randomness. The merit is "been with [Jesus] since baptism to taken"

"Lottery, past a reasonable post," is highly underrated. The randomness is there to account for the uncertainty of the objective criteria chosen ("Is it the right criteria?" "Did we measure correctly?"). Work in an escape clause in case things go horribly wrong with the ultimate "choice".

I strongly believe that this is how you solve elections, admissions, and recruitment (or, at least, get closer to an ideal solution).

It's hard to game random, is what I like about it.