^^ Not true! Many of the well-endowed schools offer tuition at any school, not just their own. Many do this because they cannot admit all the fac-brats but don’t want to deprive these families of an important part of the compensation package.
I can fill a hand right now with kids I know not at the school where parents teach.
Don’t get me wrong, when it’s a fit - and it’s easy for it to be for a kid who grew up there and whose parents embrace the culture, staying “home” can be a great choice. But insiders often recognize quickly when it will not be.