I think it is more than public speaking and small talk. As @cinnamon12 mentioned above, it has to do with their vision and values and how that has been embraced by faculty and administrators; how it has “seeped into” all parts of the school and how it is run.
Different HOS have different personalities, too. I remember that when we visited Millbrook, my D was blown away by how the Head knew a lot of details about every single kid (we had met him as we were leaving the Admissions office, and he invited us to walk across campus with him, introducing us to kids along the way). This was in mid-September. He told us that he had a self-imposed deadline for being able to greet each new freshman by name and know a bit about their background and interests by the end of every September. Not all HOS will seek to have such a personal connection to the kids, especially at a very large school, and I suppose for some kids looking at schools, it won’t matter. It did make a difference to us, and while she didn’t enroll at Millbrook, my kid was very impressed ( as was I).
Pieter Mulder, the HOS at Berkshire, also makes a point of actually knowing every student there.