If the French King was so weak , why did the South lord never become Kings themselves ? Thanks

Well, there’s a couple different answers:

1. The title of King was less important than we often think it was. 
2. The Dukes of Aquitaine married into the English royal family, producing what was known as the Angevin Empire, so it became something of a moot question.
3. There was an independent kingdom in the extreme south of France known as the Kingdom of Navarre, although it ultimately was squeezed between Spain and France as those monarchies centralized.
4. As for the rest, see here.