I agree the HOS matters a lot in setting the culture and tone of the school. Both of my kids attended a PreK - 8th grade school, and we experienced a change of leadership halfway through. By the time my kids were in 8th and 6th grades, we were done supporting the school. My older son graduated 8th grade, and we pulled my younger one out and sent him to public school for 7th and 8th grade. That new head has since been ousted after damaging the schools reputation, and spending insane amounts of money the school didn’t have to spend.
My boys will be attending two different private schools next year, and I feel good about both HOS. While the schools are quite different and one has significantly more financial strength than the other, I do feel confident in the leadership of both heads. I would have serious reservations about sending my child to a school whose HOS did not impress me, or align with my values.