There was once a kid born in San Francisco to parents who were arguably subjects of the Chinese emperor. Although they hadn't stolen any airplane (planes not yet having been invented), they were definitely here illegally, and they thumbed their nose at the Chinese Exclusion Act.
When the kid returned from a visit to the Old Country, the San Francisco Collector of Customs denied him entry. No doubt, he would have found your argument compelling. However, when the dust settled, the kid had the last laugh, 6 to 2. See Wong Kim Ark.
Wong Kim Ark did not address the Constitutional term "natural-born citizen," even though it referred back to another Supreme Court decision, Minor v. Happersett, cynwoody. Wong Kim Ark was determined to be native born. Note the lack of the term "natural-born," here. They are not the same, under the Constitution and under the law.
The reference back to Minor v. Happersett does not support your contention, either:
"At common law, with the nomenclature of which the framers of the constitution were familiar, it was never doubted that all children born in a country, of parents who were its citizens, became themselves, upon their birth, citizens also. These were natives or natural-born citizens, as distinguished from aliens or foreigners. Some authorities go further, and include as citizens children born within the jurisdiction, without reference to the citizenship of their parents. As to this class there have been doubts, but never as to the first.
Under Happersett, a natural-born citizen is correctly clarified, via understanding of this term of art going back to Bingham, to the John Jay letters, and to Vattel's The Law Of Nations, to mean born citizen without a doubt. Doubt entered the picture due to Obama being born a British citizen under The British Nationality Act of 1948, in effect at the time of his birth. He cannot be, as a state of nature, considered a born citizen of the United States, when he was born British, transmitted to him by his father.