The way I read this those parents thought they would not need ANY travel documentation for their children as long as they carried their birth certificates.

It used to be possible, until quite recently (and maybe it still is) for children to be entered into their parents' passports. My passport (issued in 1999) has a page on which to enter accompanied children. These entries would have to be made by a passport office - parents can't just write in their childrens' names themselves.