Smart kids often come away thinking they are the smartest and that they don't need to learn or understand anything else, thanks to our cheer leaders. This can lead to cognitive and behavioral fallacies, expertise in some narrow field gives themselves the idea that they are smart and qualified in domains outside of the scope of their expertise. Meeting other smart people, exposure to domains outside of one's own scope of expertise tempers that to a degree.
In retrospect the most valuable thing about my college/university experience was meeting people from different backgrounds (financial, social, educational). High school was too homogeneous.
You don't get to MIT without a combination of ability and internal drive, but for most of us, the external drive was also very beneficial. The comradery also really helped me push myself harder.
I'm in the middle of the Standford introductory compilers MOOC right now, but with a self-paced online course, the atmosphere is just very different from being surrounded by really bright and passionate people trying their best. Not everyone needs or benefits from that sort of environment, but I found it very helpful.
IHTFP
In high school, I took 6 trimesters of honors-level math at one of the "Public Ivys"[0], and my senior year of high school I was taking a full course load there plus 1/4 course load (band class) at high school. I got all As, except for a B+ in Intro to World Politics, without working very hard, driving 45 minutes each way to and from school. It was a good education, and I could have gone there on a full ride scholarship, but I still feel MIT was worth it.
At the "public ivy", I would have gotten better grades, and probably would have put more effort into personal projects, but focusing on personal projects is very different from focusing on things you have to do, and I didn't really have to focus on my school work at the "public ivy" in order to get above a 4.0 GPA. I probably would have had to have waited until after school to learn how to pick myself back up after failure.
On the other hand, I don't think I would have found some other top schools as valuable. I have a friend who graduated with a CS degree from an Ivy League school and then came to MIT. In his experience, that Ivy treated students like they already worked very hard to get into school and deserved high grades, but it seemed to him that at MIT the culture was very different, seemingly even for undergrads. What I've read about the grading curves (median GPAs, percentage graduating with highest honors) at some of the Ivys seems to match up with his observations from that one Ivy.