From what I keep hearing, you can easily outdo a university graduate if you read books on the subjects for the time of the ongoing education. While it may not be true for everybody - indeed, those who do so seem to be revered as genii or otherwise exceptionally intellectually capable - it certainly doesn't seem impossible to learn on your own even in the most complicated areas thanks to the Internet and public libraries.
Also, when it comes to learning, schools touch upon subjects that are actually important in life - human rights, day-to-day human psychology, first aid, personal finances etc. - too rarely, and such school are too far apart to make a big difference. Instead, students are forced to study subjects that often bear no significance to their afterwards life. Certainly, knowing basic physics and mathematics is very important for everybody, but anything further ought to left for advanced ("profile", as it's called in Russia) studying for students who want to learn it on their own because they're interested.
Cooperation and socialisation are another two important traits that people usually assign to studying in schools. It may be a good thing for you to learn if you got put together, against your will, with good people, it also may be a very bad thing for bright students who get stuck together with kids who are growing to be criminals already. In Russia, students aren't encouraged to spend meaningful time together: they're simply assigned to a class and have no choice other than to change it to something less appaling/better (if there're two or more same-year classes) or to change schools altogether (you can visit in other cities if you want to in Russia; which some students have to, given that their schools are sometimes kilometers away, in other towns and villages). From what I've heard, the practice is the same in most first- and second-world countries.
One of the arguments against such statements is that parents ought to hold such responsibilities, not schools - which, to me, sounds cynically uncaring. The clear trend in the mentioned parts of the world is parents spending their whole days at work to make some sort of living to support their families. Grandparents might help, but it's impossible and idiotic to ask them or make them take care for their grandchildren's mental development while the kids spend most of their days at schools; even if we assume that those grandparents will be there for them for long enough to teach them useful subjects - or even one of those, if done well enough - how useful might it be to dedicated even few more hours to learning for the growing minds? They may be very receptive, but they aren't machines of learning: they're nothing but human beings with less experience than their parents (most often), which is what adults don't keep in mind, either through not knowing it or forgetting it.
The same ideas will apply to most universities: stuck with people you don't have any choice of (or even a process to separate you into groups of people who cooperate well), spending your years learning stuff that you can most often pick up from literature of public libraries and the Internet, and having very little time of your own to spend on things you can enjoy or even build your life around. Moreover, the change from being a school child, when teachers take care of you and your mistakes might very well be easily forgiven, and a university adult, when you have to take care of yourself and your life with mistakes starting to affect your life in ways they would without the safety nets, is so sudden and impactful that it leaves many students messed up emotionally and mentally, with plenty not holding up to the pressure. I know because I was one of those students: I bit off more than I could chew and ended up failing my first year dramatically, after which I left the uni.
So, what is it that only schools or universities can give?