Stanford is a terrible single data point to use as a representation for "private schools". The amount of fundraising they do tends to approach a billion dollars per year (2009 was a bad year, they only got $640 million that year) and their endowment stands at almost $17 billion dollars.

Having a boatload of super-rich, super-giving alumni is great, and helps keep tuition costs from rising too quickly... but it is not the norm for private schools.