Birthplace would be interesting, but there’s plenty of famous/important people who are connected to a location other than their birthplace. Think sports stars who were on a team far away from their home.
And with birthplace, I think it's pretty interesting to browse the map for random tiny towns no one has heard of that happen to be the birthplaces of very famous people.
The problem that LA has is that there are a lot of very famous people you might associate with it. It seems like a nearly impossible problem to solve.
I do agree that birthplace would be a more interesting map, but I think it would be more interesting for tiny places. I think LA would suffer from the exact same problem.