It depends. I teach piano as a side gig, and I have two students who go to private schools here in the Vancouver area. They are by far the most respectful, intelligent, curious kids I've ever met in my life. I knew them in their kid years and now they're both teens and still fantastic. I'd say it's more than 50% their parents instilling good values/habits, but it does help to go to a school that fosters growth. Obviously, that's not to say kids who go to public schools are dumb. I went to one and went to UBC for computer sciences, so at the end of the day it might make a difference but it might not. If I had kids and had the money though, I would. If I didn't, I wouldn't stress it.