Surak was not entirely certain about older Spock's explanation, but he was unwilling to be inhospitable. Also an indication of a culture from a difficult place. Even the stranger is not turned out to the mercy of the land.
That's a really good point. And you made me realize there's also still room for Vulcan culture to have elements that we humans/the audience can't appreciate because we simply don't notice them. Maybe Vulcans can recognize subtle similarities in their own family members that allows them to judge by sight (or sound or smell) whether someone is a close/distant/non-relative (not speaking literally!), and there is no need to make someone recite their pedigree to prove they're part of the family.