To be clear, the usage you're describing is so rare as to be practically nonexistent. You would never use it this way if you were familiar with the word and its usage, only if you knew the word from dictionaries. Look at the rarity even for saying something like "the crew berthed the ship." You just don't say it like that.
Yes, someone "could" do it and we'd understand what they meant. But there's no reason to believe that's what happened here. Maybe "moored" did trigger an association with "birth"/"berth", but "birth" is not a mistake.
"Berthing" is also not a singular, one-time event, any ship has a multitude of berthings over time, in this context it makes no sense for there to be a "time that berthed it" as if it was some singular meaningful event, and it changes "time" into something that "berths" books...what would that mean in this metaphor that a time "berths" a book? Wouldn't other times also berth books in their ports? A time does not "berth" a book, it births a book.