It is simple jealously
I'm privately educated. I come from a VERY poor background, including regular stints without enough food, belongings being pawned etc. So there's nothing snobby about me. My parents did not pay for my schooling. Not being hungry was a real revelation to me when I first went away to school.
My child was privately educated. Neither I nor my DH were in highly paid roles. We went without and lived in a teeny house for many years. No foreign holidays. I drove a Nissan micra as old as the hills for many years. It rattled a lot.
We did that for a reason. Despite all the people that come on here and say how snobby and elitist private schools are and it's against their principles for some people to choose to pay for their kids, it's not their principles. They just don't want to make the sacrifices to give their kids a better education. They'd rather drive round in a newer car and go on foreign holidays. And that's fine, bully for them. We all have different priorities but they just cannot wait to stick the boot into people that choose differently. I preferred to give my child the best education I could in an area not blessed with wonderful schools., but goodness people get exercised about it.
Yes there were wealthy families represented at my school, and also professional families, and middle class families, and kids on bursaries, and kids of business owners from printing firms to a couple who owned a cleaning firm but both also cleaned full time themselves, to tradesmen to a carpet supplier and everything in between.
And far from one previous poster's theory that private schools teach their children to look down on other,s the reality is quite the opposite. We were regularly told how very fortunate we were and how we must be very grateful for our education and for having the means to pay for it. I work with someone who went to a far posher private school then I and our experiences of being taught humility are both very similar.
Oh and lots of kids would weekly board or part time board because they wanted to not because their parents didn't want them. What a vile thing to say.
Your SIL is behaving appallingly. It is completely inappropriate of her to punish your child because she can't or won't provide a private education for her own child.