Well, the most obvious one is the "good and wise king." I read about some Russian saying "there is no better government than a good Czar, and no worse one than a bad Czar. Unfortunately, there aren't a lot of good Czars."

One major flaw in monarchy is that usually it's hereditary. Kings are brought up knowing they will be kings, and with few exceptions, this tends to create the sort of personality you don't want running things.

Mythological kings were often picked from humble surroundings, like God picking David, or the ascension of King Arthur. I know there are many other legends of this same kind, but those were the first that came to mind. Those who wrote such legends may have understood that the wisest ruler would probably be one who knew full well what it was like not to be a king.

These kings also had powerful and wise advisors, Merlin for Arthur, and various prophets for David.

And despite all this, both kings still had some pretty hefty flaws.

To be fair to your story, history does contain cultures where the leaders were acclaimed by the tribe or people from among their own number, and could be dethroned the same way. This seemed to work pretty well, but it doesn't scale up very well to entire nations.

I'm not that tickled by how we pick our presidents. There seems to be a lot more luck to it than there is deliberation, and it's a bit weird that vice-presidents are selected by little more than the whim of a presidential candidate. Wish I had some brilliant solution, but I don't.