Every private university takes tons of public cash for research. The most prestigious and exclusive private universities take the greatest amount public research funding. If an institution wants to play the "but we're private!" card, I'd say let them, but only if it means they are not eligible for public research funding.
It's not the universities that apply for research funding, and then build basketball courts.
Researchers themselves write and apply for grants, and the grants are for the researchers. The money goes to them and their research, not the university.
For each $1 of federal research funding, the university can take a cut of as much as $0.6 owing to the fact that researchers are using university facilities and admin staff. In fact, the money itself is not even managed by the recipient researchers themselves. The university manages the funds since they use them to pay the professors, grad students, etc.
And if that research was happening at another university, it would still happen - but the benefits of being involved in it (or even being around it) would be spread out more evenly.
