I would probably imagine that the % of people that go to stanford and succeed is also related to stanford being able to only accept students they think will succeed and still have enough students. For many state schools that just is not an option if they want to accept more than 20 students. You could probably reasonably guess that 95% of the students that end up going to stanford would be successful while they were freshmen in highschool, and they probably would end up successful if they went to any university.
Nicely put, but don't stop there.... and if they didn't go to any university, it is quite likely that they'd still be successful even without a degree.
Good judgment comes from experience; experience comes from bad judgment.
A friend of mine says, the strongest indicator of future productivity is past productivity. People who do well continue to do well, and people who haven't done anything continue to do nothing. The admission process at universities is largely designed to filter down to people who have been productive throughout their earlier years, excelling beyond their peers in various respects. Whether the experience AT one of these universities has a positive effect on the students is a complicated topic. Like most things, it depends on the exact school, the exact person, and the exact environment they find themselves in. But there's a front-end selection system that heavily favors those who are already great.
(Nepotism and money notwithstanding.)
That's not to say that all talented people seek out or attend upper tier universities, or that there aren't people who show poor productivity in high school who go on to do great things later. But you have this front-end selection mechanism, reinforced by cultural biases, which tends to favor a certain sort of person. Those initial criteria are not perfect by any means, but a lot of the people out of those schools go on to do great things. As someone else pointed out, (potential) exposure to top tier research facilities and some of the best academics in the world doesn't hurt.
I don't think, however, that universities create successful or smart people. They are mere enablers and selectors. And they're not mandatory to success either.
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Having attended both a state school and a top private school as an undergraduate (albeit as a math major, not a CS major), I can tell you that, for me, there actually were huge differences between the schools. Not only did the top school offer more classes, but the teachers pushed you a lot harder, the homework was harder, the tests were harder, and I was surrounded by other students who were equally capable as me in the subject. At the state school, I took what was considered there a "senior level" class (abstract algebra) and did significantly better than the seniors even though I was a freshman at the time. At the school I transferred to, this would be considered completely normal. That was actually a pretty big deal for me personally.
All that said, math is a little different from CS. The study of math is more focused on classes and homework and teachers and peer students. I feel like the study of CS is a lot more autonomous and, like other have said, classes matter a lot less than what you do outside of class.
Certain top schools like Harvard tend to attract a truly remarkable critical mass of talented math students, and the environment is unlike anything else in the world. It cannot be replicated at state schools. At 99% of schools, for instance, Putnam Fellows are mythical super geniuses that exist in a faraway land. But at places like Harvard they're almost a dime a dozen.
Steve Jobs truly inspires me while reading this question, he didn't really need an engineering degree in college in the end. Calligraphy class was more than enough
