Something worth considering though, that just because the "Sparrows" themselves haven't existed in lore until Cersi raised them into prominence, doesn't mean they didn't exist. They were called Sparrows after the events mentioned above, yes, but the High Sparrow was inspired by the acts of those who came before and the gods themselves Westeros has always been alive with religious strife, even during the "peacetimes". Let us not forget the High Septon's Rebellion, and just because they have been beaten into submission, does not mean that the faith would just up and disband other groups, so it's possible that the Sparrows could exist as a secret society of religious fanatics, especially when you consider that their rituals are taken straight from the original andals themselves.
The concept of rebirth isn't a new one to religion, even if it's just a matter of ideologies and theological shifts in core values, so it is theoretically not unimpossible for a religious organization to exist and just change it's name.
Let us not even recover the ground that Societies, mechanically speaking, are for offering patronage, as well as gaining some benefits that come along with being a member.
And as for societies just disappearing, I have some news for you. Sure, mechanically a society can't just poof on ya, unless we've got some rather creative dev skills of course, but it IS possible for a society to "Die out" with all of it's member's having died. Just because they don't have members from the start doesn't mean that some theologist could join, or, an even better idea, potential for an event trigger for a lord to "Revive" or "Start" the sparrows like Cersi did.