Sounds like a suburban NYC school. None of them care if your kid goes. They may lose state funding, but they barely have room for all the kids anyway, so they don’t fuss about this. Kids go to private and religious schools all the time. Plus, for every smart kid that goes, dozens remain.
My kid has had annoying teachers too, but in my experience, the worst thing about public school is not the teachers, but other parents, who can be very judgemental, selfish and intolerant. Teachers generally will work with and accommodate my kid. Other parents are so fixated on their own little angels, I wish some of them would transfer out!
Perhaps it is just this one nitpicky teacher? Tho why would you want a rec from him/her anyway? Be careful your own interactions don’t poison the well for your kid there.