Showing real leadership is hard, harder than worrying about the news cycle. Real leadership is not about flash, not about showing dominance, and definitely not about looking the part publicly.
Who knows what he needs to work on in his first 3 months? Perhaps it is important to listen and understand everything before making decisions, rather than jumping to react so it looks good from a PR perspective.
Writing any culture statement is not hard, but running a process to understand the organization, set a clear direction for the future, and then solidifying values so that everyone in the organization knows is not. Writing a culture statement is also not the only thing he's done, just something that is public.
On a side note, the second you say "lavishly paid CEOs" and "he gets paid enough", you bring a level of disdain into the conversation that makes it hard to continue. We should absolutely hold CEOs of large companies to a high standard, not because of their pay, but because of their position and the responsibility they hold.