I agree with the comments about what the board wants. It is like CEOs, if the corporation need to evolve or change, there are leaders who are better suited to that. Whatever the board wants, the HOS reflects those priorities. Fundraising, public outreach, campus culture, whatever.
I only have experience with one HOS, he has been at Cate a long time, and I cannot tell if he bent to the culture of the school or the school bent to him. But they are definitely deeply intertwined.
IMO he strikes a difficult balance of being outward-facing (Public relations) and inward-facing (knowing the kids personally and re-enforcing values).