The only time I can see “subject to the jurisdiction thereof” being an issue is when a child is born in the US to non-citizens who have transferable diplomatic immunity, which is to say, in some cases, the immunity of diplomats is conferred on their resident children, and in some cases it is not. I’m not sure how it is decided when it does or does not, but if a child is born into diplomatic immunity, then you could legitimately say she was never subject to the jurisdiction of the US.
Nations have the power to waive diplomatic immunity for citizens abroad who have it, and have in a few cases, when the children of diplomats have committed heinous crimes-- there was a case once, where the son of an ambassador who was accused of rape. The governor of the state where the consulate was located got the nation to waive diplomatic immunity, but after the trial the sentence was carried out in the rapist’s home country, as part of the whole deal. I’m not finding the case now, but I remember reading about it-- I think it may have been New York, because I seem to remember reading about it in The Village Voice.
Anyway, that makes me wonder if a country could retroactively waive someone’s diplomatic immunity if the person wanted to claim US birthright.
Here’s what I’m waiting for, though: the test case of the foundling.
There’s a law that if a small child is found within US borders, who, after reasonable efforts to determine its origins are exhausted, and its parents still cannot be found, so that there is no way to know where the child was born, is determined by a physician to be under the age of 5, then the child is presumed to have been born here. This is for purposes of receiving services, becoming eligible for adoption, etc.
As far as I know, no one whose citizenship came under such circumstances has ever run for a high office, but since the presidency is the only one that requires natural-born citizenship, I suppose a foundling could always undergo naturalization if the constituency was unhappy.