“There are only two categories of US citizens, natural-born and naturalized.”
Perkins v Elg(1939) seems to support your argument.
“If he was born elsewhere, the issue gets complicated, but he may still be eligible.”
I can’t see how. His mother wasn’t old enough under the law in effect in 1961. Barry Obama has probably never been a US citizen, and is likely subject to deportation.
I'm not an attorney, but this seems unlikely to me. I seriously doubt the intent of the law was to produce a citizenship trap for children of young married American women.
I do know the present version of the law requires the American parent to have lived in the USA for 2 of the previous 5 years.