The author fails in his second sentence. All actions of the US government are governed by laws. Those laws laws come from acts of Congress. Article 1 Section 8 of the Constitution enumerate Congress with the “Rules of Naturalization”.
The scope of those rules are not limited. The rules include who needs to be naturalized and the rules include who does not need to be naturalized.
The current expression of those rules are listed in Title 8 section 1401. Even those citizens who are born in the US are citizens at birth (Naturally born as citizens) are defined by that law.
Rules of naturalization have nothing to do with rules governing natural born citizens, which is defined by the standards of natural law. For example, from the book "Laws of Nations" written in 1758:
"The citizens are the members of the civil society; bound to this society by certain duties, and subject to its authority, they equally participate in its advantages. The natives, or natural-born citizens, are those born in the country, of parents who are citizens. As the society cannot exist and perpetuate itself otherwise than by the children of the citizens, those children naturally follow the condition of their fathers, and succeed to all their rights."
From the point of view of the founding fathers, Ted Cruz would be Canadian, because he was born in Canada to a father of Canadian citizenship. He would not be a "natural born citizen."
If you took out that particular section of law, those persons would still be citizens under the constitution. Their citizenship doesn't depend on the statute. Cruz's citizenship depends on the statute.
Congress certainly has the power to define who is a citizen, that comes with the power to make rules for naturalization.
The SCOTUS cases cited above say that citizenship that depends on a statute is naturalized citizenship. That's what they say. You may disagree with that conclusion, but it's the conclusion of SCOTUS.
Even those citizens who are born in the US are citizens at birth (Naturally born as citizens) are defined by that law.
From the Article, quoting the supreme court:
[O]ur law in this area follows English concepts with an acceptance of the jus soli, that is, that the place of birth governs citizenship status except as modified by statute