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.