This is a tricky one to me. At a very basic level, no school should hire a head who can’t promote its values and advance its goals. So if a HOS is saying things that just seem off to you, it may be that the school has the wrong values for you. And while I strongly believe that leadership sets the tone, I’d be wary of any school that had became dependent on its head to maintain that tone. You and your kid are going to have little interaction with the head compared to tons of other adults at the school.
Otoh, when a school hires a head, it may have have different priorities than a parent and these may not translate into the same things you are looking at/for. For most parents, you will see the head at public functions (from revisits to sporting events). If you are well-off, you may see them with advancement officers. And if your kid gets into a particularly complicated mess, you may get to know them. But for most of us, we are rating them as public speakers and chit-chatters.
As in all things, I think it depends. I’d think carefully about what you liked or did not and try to sort through that.