In my experience (former PhD student and instructor and my spouse is a full-time faculty and program director), if you're a tenured faculty member at a R1 university, then grant writing is your job. You're expected to be self-funding and the only way to do this is to grind out grants and pass the work onto graduate students and staff. And, you pay a ~50% tax on your grant award as 'overhead' to the university, so you have a lot less money to work with than the sticker value that gets awarded.
At the university level, arguably the second most important person in the administration is the President of the Foundation and they don't even technically work for the university as the foundation is a separate legal entity. This person has a massive private army of fund-raising staff that shakes the alumni money tree looking for gifts large and small.
But, that doesn't mean that universities don't also have money chasers. My former uni had a special division in the recruiting/admission office that focused on high-value future alumni, aka, children of rich and famous parents. These kids were brought in for tailored events, private meetings with faculty and admins, and received a general red carpet treatment to try and lure them to the university.
Related, Malcom Gladwell has done some great work dissecting the money making machines of the administrator class who go after $100M+ gifts (far more money than 99.99% of researchers could acquire in a lifetime of grants). And, how this gift giving continues to pile on at elite schools while starving everyone else of gifts.
http://revisionisthistory.com/episodes/06-my-little-hundred-...