If you're a high earner, your kids may well pay full sticker price. The whole point of price discrimination is to let poor kids in for less while soaking as much as possible from rich kids.
After doing research I am finding that grants and scholarships are less common than I originally believed.
I did not go to the most prestigious school I was accepted to. I went to a less prestigious school that offered a much better deal. I suspect many students can receive merit based aid by selecting a lower tier school.