Role for Sortition in Selecting the Executive?

Most of the discussion on this site has understandably been focused on the legislative function. What about the executive branch, whether a single president, or full executive branch?

In ancient Athens, we know, the executive magistrates (typically operating in boards of ten) were selected by lot, as was the Athenian president (though this office was primarily symbolic and rotated daily.) We also know that sortition was used as a step in a convoluted process of selecting executives in several medieval Italian City Republics.

Is there a beneficial role for sortition in selecting government executives in a modern democracy?

Some of the ideas that I have come across include:

1. Having an allotted body interview, hire, and fire the state executive, similar to the way that many city councils appoint a city manager.

2. Having a pool of voters selected by lot elect the executive, as a way of overcoming rational voter ignorance in  a mass election.

3. Selecting an executive by lot from among a pool of candidates who achieve a given threshold in a popular election.

Thoughts?