That's doubly false :)

First, all social networks that preceded it (LiveJournal and MySpace were big amongst my peers) had open sign-ups.

Second, Facebook became popular because the sign-up was not open. You needed to have an .edu email address to sign up. This was a feature, as it kept parents and younger siblings off the networks. Having a FB account was a literal right of passage, something you can only get once you go to college. It spread because of that; those were the great days of FB.