Is the desire for leadership connected to our own vanity?

In some instances, yes. In other instances, no. There are those who bedeck themselves in regalia and are always at the front of the stage saying "look and me and how important I am." There are those who will be constantly checking social media and the like to see what is being said about them.

And then there are those who undertake a leadership role, in part, for the good of others. People who join an institution because they think its ethos can make a difference. People who inspire tremendous respect and loyalty in their staff through encouragement, decency and a sense of fair-play. These people haven't taken on their roles solely with a view to their own betterment and improved remuneration. For them, vanity doesn't come into it!