Why do some fellowships require "leadership experience"?
Precisely because we cooperate a lot, we need good leadership. A good leader is someone who can bring the best of everyone in the group. For example, encouraging everyone to voice their ideas, assessing which ones have actual value, but without making the group pursue all the wild geese. A good leader should also be able to recognise the strong points in the members of the team and assign tasks accordingly.

I accept that everyone here should be motivated, but it is important for the group to keep it up. No matter how eager I am, if my professor were to start giving me contradictory orders, unreasonable workloads, or dismissing my ideas without explaining why, I can tell you, I would not remain that motivated after four years. After all, I can be motivated to work "in the grand scheme of sciency things", but not necessarily in the particular group I am in.