Colleges certainly aren't altruistic.
For the most part colleges, and even more so, research universities, want to be "better than each other". The competition isn't for money so much as for its own sake.
It is easy to find out what colleges think is worth bragging about: visit any of their homepages.
It is important, as patio11's quote suggests, to remember that the interests of different groups -- undergraduates, graduate students, faculty, administration, alumni -- are not necessarily aligned. When I was an undergrad at Rice, in addition to the student newspaper we had "Rice News", rather similar in content to the "news" on university homepages. I regarded this as propaganda and sneered at the "faculty newspaper"; now that I am on a university faculty, I realize that the faculty don't read these newspapers either.
From an administrative level, most everything costs money. You can build new buildings, fancy new dorms, start new programs, poach famous faculty, etc. And so they think about money a lot. Departments don't have to think about where the money comes from, but they are always fighting to get a piece of it.
Professors want good students to teach, time to do research, and to be left alone by the administration. University administrators always have pet projects (at South Carolina it's the "Carolina Core") and it's rare to find a professor who gives a damn about any of them.
You mention "a compassionate job of preparing eighteen year olds for adult life". But "adult life" means very different things to different people, and many professors, certainly including myself, are extreme outliers. So I personally stick to teaching my students math, and trying to get them excited about it -- that is what I am good at.