This is how it works in recent decades for US child support, but it is not how it worked for making an illegitimate child a UK subject.
Read the 1948 BNA. It only passes UK citizenship to legitimate children, not any children whose birth certificates bear the name of a UK citizen. Big difference.
The US constitution and courts cannot declare Barry to be a dual citizen at birth just because a UK subject is on his (alleged) BC. Only UK law could do that and it explicitly excludes illegitimate children of UK subjects.
Well considering there is a divorce decree for Ann and Obama Sr. that is gonna be a tough row to hoe.
A court in the UK would have to declare Obama illegitimate and officially negate the inherent citizenship claim. This isn’t something that can be done in a U.S. courtroom during an eligibility hearing. The physical evidence that is available right here and now says the father was legitimate and that his son was born with a natural birthright to British citizenship. Obama’s Fight the Smears page works against him, plus the citations in Shanks v. Dupont, Inglis v. Sailors Snug Harbor, The Venus, Minor and Wong Kim Ark, several recognizing the distinction of British citizenship for the children born in the U.S. of parents who adhered to their British citizenship, as distinguished from natural-born citizenship.