Does the whim of a private company really get to determine if you get a scholarship or not - and thus profoundly influence your future career? Or are these universities really implicit public institutions?
I don't think Stanford university is purely private in the same way that, say, the Coca-cola company is private.
The scholarships may be privately funded, but 84% of Stanford research funding in 2012 was received "directly or indirectly from the Federal Government". That's a quarter of the University's overall income, according to its annual report, before you start accounting for tax breaks. I don't doubt the government gets a return on that strings-attached investment, but welfare has positive externalities too...
That's a great point, and you could also ask how "public" universities are as well. UC Berkeley now gets about 12% of its budget from the state.
Its a little difficult to make direct comparisons to Stanford, as Stanford's numbers will include a medical research university while Berkeley and UCSF are considered separate campuses, and Med centers have huge budgets. But the state financial support for UC Berkeley has dropped dramatically (around 12%).
It may be different for the smaller state schools that are not research universities. But if you line it up, private and public universities research universities are more similar than different in their sources of funding (grants, private donations, endowment income, etc). The things people believe distinguish these universities from private ones (tuition, state support) are actually a relatively minor part of the budget.