It’s not even relevant why the child might “want” to wear a kippah serugah; if the school told him not to, he shouldn’t.
If the type of kippah someone wears was really completely immaterial (pardon the pun), we wouldn’t be having this discussion, and the story never would have happened. You won’t hear this story about a four section black velvet yarmulke vs. a six section black velvet yarmulke.
Whether you like it or not, yarmulke styles do represent hashkafos.
What the father was doing was pointing to the fact that there’s no inherent value to a particular style, to which Rav Shteinman agreed, but Rav Shteinman pointed out the father’s hypocrisy in that he obviously personally wanted to be able to superficially represent his hashkafos (otherwise it wouldn’t be worth disobeying the school policy and getting expelled) but he didn’t think the school had a right to superficially represent its hashkafos through a policy regulated what it’s students can wear.