That’s easy to say now, but Orkut (another Google social network, mostly used in Brazil) also had a beta invite system… And that helped it grow tremendously. The secrecy and “status” of getting invited made people go wild - they would even sell invites.
The strategy can work. It’s just very timing sensitive.
I’m not sure if I’m mis-remembering, but I believe my first Gmail account was by invitation. It was pretty much just an email account back then