Maybe cause Square wanted you to feel like a part of the adventure and he was your own character who originally (in game) didn't have a default name. Its much like Tidus in Final Fantasy X.
For some reason I look at him as leader cause of how I set up my party in 1,3, and 5. Knight, Black Mage, White Mage, Thief. I support the knight as leader cause he is the basic strength behind the party. The other characters are great but the knight is a much more real role to play.
Also having him as the character who travels around helps. I think you can switch who it is that walks around towns and the world map. Although I'm not sure. It's nice to have him as leader cause while other characters has their role, Bartz looks after everyone and makes the hard decisions. I'm sure if someone else was leader we would have a topic asking why the someone else is leader. :P