More prestigious private universities use a lot of government funding to fund widely cited research which is what makes them prestigious.
One can reasonably argue that such protected categories are necessary for a just and fair society, but let's be clear about what it is we're advocating.
Just a correction, the unversities are NOT the ones that apply for, and receive funding from e.g. the NIH.
It is individual researchers that apply, and receive, funding. And this money goes towards their salary and research. No funding, no salary, no reseacher.
Universities don't themselves receive government funding.
Each university has its own F&A rate, which can be as much as 60% of received federal funds [2]. This rate has historically trended upward.
An example of funds allocation for a typical small NSF grant: https://austinhenley.com/blog/grantbudget.html
[1] https://files.eric.ed.gov/fulltext/ED517263.pdf
[2] https://financeandbusiness.ucdavis.edu/finance/costing-polic...
This is one reason I'm leaving academia – if I raise money outside of academia, I actually get to keep it.