All politics is local. I don't know the make up of the school district, but if there were a Jewish or Hindu parent they would certainly have standing to file for an injunction. And the district's lawyer would have to charge the district to defend it the in a case he would lose. So not playing the song in the first place is a smart move.
Absolutely. A private school is like a private person. There is no law forcing a child to go to a private school or the parents to pay for it. But public education is both a right and a legal requirement. So the school is required by the first amendment not to push a particular religious philosophy.