How about the law at that time that the child’s citizenship followed that of the custodial parent, in this case, his mother?
What makes you think that was the law at the time?
That would be a terrific story, except it's just that - a story.
Since AT LEAST 1898, the law of the land on birthright citizenship has been the Supreme Court case United States v. Wong Kim Ark, 169 U.S. 649 (1898). In it, the Court holds that anyone born in the US subject to the jurisdiction thereof, irrespective of the citizenship status of his/her parents, is a citizens of the United States.
They are not citizens because of statute, but because of the 14A. No statute, like your ridiculously absurd "law at the time that the child's citizenship followed that of the custodial parent" can disenfranchise a person from a right that they enjoy via Constitutional law.
If you are a citizen at birth, you cannot lose your citizenship, nor can your parents by design or neglect, renounce your citizenship either. There is ample case law since Ark that says just that.
His mother was not old enough to give him US citizenship.