Public anything is a mistake because you can never run anything in a way that someone won't have good reason to dislike, and by making it the defacto instance you are often forcing people to participate in your bad ideas if they want to participate in something similar at all.

There should be no public sports. Anyone who wants to play soccer while in high school should join a private league. Then that private league can weigh the interests of different people who want to participate and set their own standards. If they choose ones that don't serve the majority someone else will make a league. It doesn't need to be a political matter. Remember that politics is violence. It's just organized violence. Violence shouldn't be used to get women to play sports with men if they were to want to only play sports with other women.

This also applies to public education more broadly on a lot of subjects. We shouldn't have publicly funded telling of distorted historical narratives. We shouldn't have publicly funded/selected view on legal philosophy (civics). We shouldn't have government selection of how language is used. We shouldn't even have the government select the order, selection, and emphases of which mathematical concepts are taught, because it's just likely to suck and not keep up with the times compared to a more competitive selection. Who decides that initial exposure to linear algebra concepts aren't important in elementary school? Or that trig comes before polynomial calculus? Some of these choises are just stupid.