American law supercedes British law within America jurisdiction. This isn’t difficult. Yes, you’re correct that the British issue a claim on everyone born to British nationals. I agree, this makes Obama not a natural born citizen even if born in America.
But here’s the rub. Concerning American citizenship - born in America, you’re an American citizen. Once again - you’re conflating two different arguments. It’s not just ‘what do Americans’ have claims on them, it works the other way too, ‘what does America claim’. America claims that those born on American soil have American citizenship. They can choose - by not living in America, by taking up residency abroad and they can choose to turn in their citizenship. But the point is that they possess this citizenship which may be exercised if they want to. America doesn’t strip minors of their citizenship unless they make the decision to drop it and the other particulars apply.
Why? Because the United States understands that the desires of the minor might not align with those of the parents.
There has never been such a ruling. The closest was U.S. v. Wong Kim Ark which ruled that those born in the U.S. to permanently domiciled aliens were U.S. citizens at birth.
Note, though, that Gray declined to specifically rule Ark a 'natural born citizen.'