By what logic would a person enroll their child into a private institute that enforces a non-arbitrary code that they disagree with?

I see no problem with a private Catholic school saying “If you’re not Catholic and have an issue with what we enforce here (e.g. dress codes), then you should enroll your children here.” I imagine that’s what they all would say.

If I didn’t make it clear, this applies appropriately to private institutions, meaning “you are enrolling here on your own conscious accord.”

If a person has any issues with how a private academic institute operates, then they are not mandated to attend it. Furthermore there is a difference between blatant discrimination (“No non-Christians allowed”) vs. "there is a particular culture that is being observed here; we aren’t going to bar anyone from attending this establishment, but it’s illogical to expect us to bend to your will because you are in opposition to what we practice upon you joining."