A player has never become a better leader or player by having a letter stitched onto his jersey.
Not by the simple act of having the letter on their jersey, but leadership is only partially innate. Not to say that it isn't at all, but it's just as much a skill that has to be worked at and developed. Having the letter gives a player more agency to work on his leadership and implement it... a better opportunity to become a better leader, if you will.
There's also a difference between being *a* leader and being *the* leader. Anyone whose ever moved from being a leader under the leadership of others to being a leader over the leadership of others knows this.
That being said, I don't necessarily believe that having a single captain is an absolute necessity. The best team I've ever been a part of had myself and one other person at the top, with equal authority overall and the specific responsibilities being divided up equally. There's more than one leadership model that can work.