The U.S. State Department disagrees with you. On the first point, they do. But not on the second, they agree that there are residency requirements for the US citizen parent(s).
I wish they would refer to section of the US code rather than the Act that generated it. Makes looking it up easier....ah, here it is. 8 USC 1101 (definitions). Yes it says what the manual indicates.
I think it's a matter of interpretation. And in general doesn't matter anyway, since naturalized citizens and other citizens have exactly the same rights. The other seciton of the law refers to "citizens at birth" it is true, but the law was passed under Congress power to define a uniform rule of naturalization. So how can persons whose citizenship depends solely on such a law be anything but naturalized? The part you quote from the state department is not the law, just their interpretation of it.
The State Department, in the same Manual says (in the section immediately preceeding the one you quoted:
7 FAM 1131.6-2 Eligibility for Presidency
(TL:CON-68; 04-01-1998)
a. It has never been determined definitively by a court whether a person who acquired U.S. citizenship by birth abroad to U.S. citizens is a natural-born citizen within the meaning of Article II of the Constitution and, therefore, eligible for the Presidency.
b. Section 1, Article II, of the Constitution states, in relevant part that “No Person except a natural born Citizen...shall be eligible for the Office of President;”
c. The Constitution does not define "natural born". The “Act to establish an Uniform Rule of Naturalization”, enacted March 26, 1790, (1 Stat. 103,104) provided that, “...the children of citizens of the United States, that may be born ... out of the limits of the United States, shall be considered as natural born citizens: Provided that the right of citizenship shall not descend to persons whose fathers have never been resident in the United States.”
d. This statute is no longer operative, however, and its formula is not included in modern nationality statutes. In any event, the fact that someone is a natural born citizen pursuant to a statute does not necessarily imply that he or she is such a citizen for Constitutional purposes. The law which replaced the 1790 act, in 1795, said the same thing, without the two words "natural born". And so the law remains today, that a child of US citizen parents, born outside the US, is still a US citizen.
What is factual is that the Founders and authors of the 14th did not intend for anyone with split loyalties or split obligations to qualify for president.
They did not intend for those with dual citizenship to be qualified for president.
I wish they had written it much more specifically.
None of this really matters any way. No agency of government is going to remove him just because he isn't constitutionally eligible. The point should be aimed at the States passing laws that keep him or anyone else from being on their ballots in the future if that person doesn’t produce an official long form BC showing he was born in the U.S.