I don't think it's really possible to be neutral about culture war issues, sort of by definition, especially when many of the wars are about schools themselves. A school is part of American culture, it can't somehow choose not to participate. To use a historical example, there was a culture war over teaching evolution in schools. A teacher at the time could describe the history and facts of the controversy without stating their opinion on it, but when the students leave that classroom, they are still in a school that either teaches evolution or doesn't. By existing and having a curriculum, the school takes a position in the culture war.
Imagine if your country held a referendum on a certain issue, and said "if more than 50% of citizens vote yes, the law passes, otherwise it fails". You could have a genuinely neutral opinion on the issue ("I don't care which side wins"), but you cannot take a neutral action, because not voting makes the law less likely to pass. This is unfair, and you can be mad about it, but the only way to be "neutral" would be to leave the country and renounce your citizenship so that you are no longer part of the equation. Similarly, in a culture war, the only way to be neutral would be to not participate in the culture in question, which is even more difficult than leaving a country.
Let me give one more example: whether or not to tip your server in a restaurant. Tipping 10% instead of 20% is not neutral. Tipping 50% of the time you go to a restaurant is not neutral. You could say "I don't wish to take a position, so I will no longer go to restaurants", but I'm confident the anti-tipping side would claim you as one of their own. Again it feels unfair, but it's just how culture works.