That is the real kicker - their fellow students and maybe some of their professors. The alumni network might kick in for a very very small number of graduates, but the friendships that develop during 4 years in college generally last for the rest of a persons life and influence both personal and professional aspects for better or worse.
There is also a herd mentality in any small community - surround yourself with party animals and a large percentage of future drop-outs in your freshman dorm and it will up the chances of your own failure. Surround yourself with a large percentage of people who take the academic side of college seriously and your chances increase. I saw it in high schools I used to visit professionally (and the one I attended), and the same for the hierarchy of colleges.
As for the piece of paper - when I was hiring and reviewing resumes, I certainly noticed the college name but it had no effect on the final decisions - might get someone into an interview but never got them a job.
I've always believed that you get out of a college education what you put into it. Most universities have all the same books in their libraries, offer almost all of the same majors and there are good professors everywhere. Back in the dark ages, when I went to college and grad school, I frequently found that some of the tenured professors with the greatest reputation in their fields were among the least effective teachers, while TA's and Asst Profs, working diligently towards the promised land to become tenured professors, were among the best teachers.
But that's not to say that there aren't inherent advantages to top colleges and universities like Stanford, MIT and the Ivies. Chief among them is the research opportunities available from both government and private grants in engineering, science, mathematics, the arts and countless other disciplines. One of the reasons these schools are consistently among the top colleges in the country is because of their ability to raise funding in terms of donations and research grants. A second advantage, often tied to a school's alumni, is their ability to offer quality summer internships to their students aligned with those students academic major.
My son attended a very good college to study mechanical engineering. While there he helped develop a technology under a research grant from Ford Motor Company and he secured a summer internship with General Electric's R&D Division. In addition to adding some substance to his resume, the experiences connected him directly to two Fortune 100 companies (He ended up working for GE after graduation).