I believe this is exactly the US legal position. While other countries may recognize or grant citizenship to whomever they wish, we don't pay any attention to it. It doesn't change the person's position under US law.
A Supreme Court case way back in the 19th has some bearing. A young man born in this country to Chinese non-citizen parents (Chinese immigrants weren't eligible for naturalization at the time) sued to have his citizenship recognized.
One of the dissenting justices pointed out that if we recognized him as a citizen despite his ineligible for citizenship parents, we would be opening up the presidency to "coolies," thus pretty clearly indicating this particular man was "natural born" despite his parentage.
I don't believe the other justices specifically addressed this concern in their opinions, thus implying that it was indeed a possibility and they weren't too concerned about it.
At any rate, I think we are in agreement as to the position of the government. It holds that term "natural born citizen" for presidential eligibility purposes means "born in the US" even if the birth is to a non-citizen mother who is vacationing in the US.
-- One of the dissenting justices pointed out that if we recognized him as a citizen despite his ineligible for citizenship parents, we would be opening up the presidency to "coolies," thus pretty clearly indicating this particular man was "natural born" despite his parentage. --
That's the dissent's take on what the majority ruled. It notes that a "citizenship is by birth location only" rule cuts in both directions:
If the conclusion of the majority opinion is correct, then the children of citizens of the United States, who have been born abroad since July 28, 1868, when the amendment was declared ratified, were and are aliens, unless they have or shall, on attaining majority, become citizens by naturalization in the United States; and no statutory provision to the contrary is of any force or effect.
The logic that statutes would have no effect is based on the majority holding the 14th amendment to the Constitution mandating citizenship by birth location, and the constitution cannot be overridden by statute. Under this logic, not only in McCain not eligible to be president, he is not eligible to be a Senator because he is not a naturalized citizen. This how the dissent in Wong Kim Ark characterizes the majority holding.
The statement by the dissent that touches on eligibility to be president is itself short on analysis:
Considering the circumstances surrounding the framing of the constitution, I submit that it is unreasonable to conclude that 'natural born citizen' applied to everybody born within the geographical tract known as the United States, irrespective of circumstances; and that the children of foreigners, happening to be born to them while passing through the country, whether of royal parentage or not, or whether of the Mongolian, Malay, or other race, were eligible to the presidency, while children of our citizens, born abroad, were not.
I also note that Wong Kim Ark marked a change in the treatment of persons born to foreigners who happened to be located in the US. In other words, the statement that the policy today is the same as it was at the founding is incorrect.
See too Minor v. Happersett, 88 U.S. 162 (1874), which discusses but (because there was no need to) also does not resolve the meaning of "natural born citizen." The issue in both cases was simple citizenship, not eligibility to be president.
I assume that we have irreconcilably different takes on the meaning of the Wong Kim Ark case, as well as on whether or not it was correctly decided. Again, my observation was that Congress has roughly adopted the view of the majority in Wong Kim Ark, by concluding (or assuming without debate) that a dual citizen can be eligible to be president of the US. My beef is with Congress, as I see it as derelict in its duty. This is just one more example.
U.S. law does not mention dual nationality or require a person to choose one citizenship or another. ...
The U.S. Government recognizes that dual nationality exists but does not encourage it as a matter of policy because of the problems it may cause. Claims of other countries on dual national U.S. citizens may conflict with U.S. law, and dual nationality may limit U.S. Government efforts to assist citizens abroad. The country where a dual national is located generally has a stronger claim to that person's allegiance.
However, dual nationals owe allegiance to both the United States and the foreign country. They are required to obey the laws of both countries. Either country has the right to enforce its laws, particularly if the person later travels there.
However, notice that owing allegiance to more than one country does not, according to Congress, interfere with constitutional eligibility to be president of the United States. Born on US soil (except to a foreign diplomat)? Eligible to be president, see Wong Kim Ark and 14th amendment.