I want to go on the record on the side of "Yes, people without kids are absolutely capable of caring about education."
But I also wanted to offer a correction:
Is this limited to people with kids going to that specific school?
School boards are for the school district, which is obviously composed of many elementary schools, junior highs, high schools. Without speaking for every school district in the country, I would expect that school board members would need to be residents of the district.