Be careful with that private school generalization. I send my kids to private school (on my dime). For all my kids, I would say on average about 40% of each class has kids there on scholarship - some classes more, some less, with scholarship being needs-based, performance-based, or both. So the implied "private school kids come from wealth" generalization isn't always true.
Private school kids being always rich was not implied. A skew towards wealth in private schools however, is.
Which is clearly true.
I mean, I'm married to a lady who went to a private school. I wish she was rich.
The point was, adding concessions to a system often exacerbates the gap between poor and everyone else, which is not what people would necessarily expect.