I disagree. That is why people pay so much money to get into prestigious undergrads since they believe it will lead to access into prestigious grad programs, which in turn could lead to better-paying jobs.
It used to be recommended that you move to a different program for your PhD- if you have ever read Richard Feynman's various bios, they told him to leave MIT and go to Princeton (I think that is where he was, LOL). That isn't necessarily true today.
As I mentioned, obviously you CAN get into prestigious grad schools coming from lesser known undergrads. It's much harder. How much harder is difficult to say.
Law and medical schools go on lots of factors for acceptance- where you went to college is a big one; GPA, MCAT/LSAT scores, personal essay, resume/activities, etc. Many people at Harvard Law also came from the Ivies for undergrad. Not all of them, but there were not alot of people from no-name schools either.