Pretty sure Facebook took off because they required you to have a .edu address, and even then, when it first launched it wasn’t just any .edu, the rolllout was slow. I remember people enrolling in 2-year schools when the regex matched on *.edu just so they could get an account.
Facebook hit the seam of internet 2.0: after the .com crash with a bunch of kids who grew up on AIM/ICQ/whatever and all these kids wanted to keep up with their friends at various colleges.
They indeed just got lucky.
The .edu constraint also played perfectly to the most important requirement of a new social network: Your parents must not have accounts there.