“.....Why should this one be different? The information added is crucial to the mythos of birth and parentage. It’s simply not good enough IMO to have it appear UNDATED AND ANONYMOUSLY.”
If this is the only document in that immigration file in which the kids name appears, then it is odd.
You really would know if the name appears more than once by now...if you had looked through the file to which I provided the link, several times. There’s nothing like seeing it for yourself.