This is bad news. All this while I too had thought that's how we do it: log in to our Google account on a new phone, download Google Authenticator, and we'd be up and running again. Apparently it doesn't work like that!
I'll bet many others aren't aware of the hassle they'd go through should they have any sort of issue with Google Authenticator. This post should have more visibility so as to ensure they'd know about this.