> One of the primary responsibilities of a leader of a company is public relations

Coordinating the strategy, yes.

Being a public face, also yes.

Participating in drafting the most important, and approving any important, releases? Sure.

Actually working the social media accounts on a daily basis while burning the midnight oil? Not so much. If Twitter PR is that important and Musk is so indispensable running it personally rather than delegating (which probably isn't the case, but for the sake of argument...), than he needs to offload some of the other stuff to a COO or co-CEO.