It can't go back that far, because the Progenitors' whole motivation is that there was no other intelligence in the galaxy at the time, so they arranged to seed multiple worlds to gradually evolve intelligence billions of years after they were gone. They were all alone in the galaxy, so there was no one for them to preserve. They're the opposite of preservers, because they're not keeping something around that already exists, they're creating something that doesn't.
See, this is why it's so weird that people want to associate the Preservers with the Progenitors. Metatextually, they're both attempts to explain humanoid aliens, but in-universe, they have nothing even remotely in common and indeed are opposites in many ways.
I also think it's ridiculous to believe any cultural concept, institution, or tradition could possibly endure for billions of years. Even millions is pushing it.