Should only people with school-age children be allowed to vote on school board members? This inclines me towards yes, although I understand the arguments against.
This is an obvious no. Apart from the issue of taxation without representation, not having children does not disqualify your opinion on education. There are plenty of taxpayers who are interested in ensuring that children get a good education at a reasonable cost who don't have kids.
We also do not live in a democracy but a representative democracy. Very big difference. Having the elected legislature or mayor choose the school board is perfectly in line with a representative democracy.
I live in a jurisdiction with elected schoolboards as well, and I find the notion odd and somewhat anachronistic. It seems only natural to me that public schools should be handled by the municipality's government, or maybe the provincial one directly. (The provinces have, in practice, assumed much of the responsibility anyway. I understand it's the same in the USA with state laws, and state funding, being very involved in public education.)
Even municipal elections are marginal enough that many people don't take them seriously here. No one knows anything about the schoolboard elections. I have to actively research quite a bit just to find out anything about the candidates. I often abstain out of ignorance despite trying to inform myself!
At least in California, school districts are independent from cities, and while most are wholy within one county, some aren't. The school district is the municipality. California municipal districts for the most part have elected boards, although the elections don't have to be run if there aren't more candidates than seats.
Pay attention when a California mayoral candidate says they're going to fix the schools. That's a sign that they either don't understand how schools in California work, or they think you don't. School districts aren't subordinate to cities, so mayors have no power over them. With a slight caveat that because San Francisco is a unified city and county, the mayor of SF is also on the SF county board, and counties have some power over school districts, but not a whole lot.
I have had similar questions about my native state of Florida. Florida, like many retiree magnets, suffers from having retiree immigrants who vote against the interests of those who still need schools, roads, and other infrastructure. I half-joke that we should amend the state constitution to restrict the franchise to those with skin in the game.
Obviously there would be childless voters who would have the best interest of students in mind when choosing school board members. For example, couples planning on having children might want to be able to influence the election.
Of course, a board elected by those with children could also be allowed to appoint childless advocates as well.
It seems like the democratic equivalent of micromanagement.
Those numbers are all too low, because the last category has substantial overlap with the first two.