Is the desire for leadership connected to our own vanity?

A little bit, yes. Leaders are often on spotlight, in front of cameras and in the media. They are important persons that have special mission to accomplish. All this can be tempting for self-realization and boosting ego. Desire to be listened, important and a person whom always other people go to for advice/solution is normal. We are all humans with our egos which we have to have under control and make to serve us, not destroy us.

Yes, I can say that it can be connected to vanity to some extant, but again; there are people who are not vain at all and choose leadership as a tool to make things better. So it depends.