There are 2 reasons a king gets executed. Suspected of being evil and being “worthless” because 4/4 times he investigated a BD.
The latter is not a reason the king should be executed, but unfortunately it is a lot.
I would never find a king that doesn’t find leads himself worthless. IMO opinion I don’t care what the king finds. A good king that I want to keep around isn’t somebody who can get lucky and find unseen at night. It’s the king that is talking to people all day. Asking for leads, getting people fired up, collecting and putting together the pieces of info he collects in whispers. Delegating tasks to his subjects to try to win the game.
If you have a king who is talking and working and thinking and getting everybody involved, but not finding unseen at night, and not executing somebody every day just because somebody has a hunch, he’s the best kind of king, don’t kill him. (Unless he’s evil. Then kill him)