Its not like those schools are demonstrably better its that those schools are exclusive and provide social connections to elite social classes. The whole point of them existing is to be exclusive more than it is to educate.
Global stature is virtually impossible to attain, an unrealistic goal. The world is actually a small place, and global stature is a reasonable goal within one's domain of interest. Both views are indisputably true. One tends to believe the view held by people around you. While it's easy to become a student at an elite school, it's hard to become a professor at an elite school. These are people who have "cracked the code" and naturally see the world as a small place in which one can excel. Most students will never "crack this code", but those in direct contact with good role models are more likely to do so.