Nah, not Superman. [He really admires him--to the point of stupid fanboy behavior--but he never came across as the head of the League except for maybe at press conferences.] In my opinion? It's Martian Manhunter and Batman together. They seem to be the ones with it all under control.
And you are a far braver soul than I. I was scared to even touch anything when I was in the BatCave, let alone ask questions. Wasn't allowed to touch things on the WatchTower either.