And if a school needs to be building its endowment or managing faculty issues, and the HOS isn’t doing that (but is very present on campus), that’s a problem too. Not for parents and students, but for the future of the institution.
To be clear, I didn’t mean that the role is only public speaking and public appearances. But that is the part of the role parents will see.
Based on responses here, trustees should weight parent sentiment toward a head highly.