Yes this is true. You don't have to be in charge, but then you are not doing any of the fun stuff either. The lead has the final say and will take input from game designers. They will not shape how the actually game works but rather how smaller parts of a game will work.
An example, one game designer can design how the GUI will work and interact with the user.
Sometimes they do make the story but not always, They may come up with the main Idea but not all the interlinking events. There is a lot of writting that goes into a game including non story text. Every character in halo from grunt to spartin has text written for their in game voice acting. One person would not write all of that and keep their sanity.