Congress only has the power to create uniform rules of naturalization. The law you link to is a uniform rule of naturalization. It can be nothing else. Hope this helps.
The Supreme Court reaffirmed the primacy of the 14th Amendment in its decision in Schneider v Rusk in 1964:
“We start from the premise that the rights of citizenship of the native born and of the naturalized person are of the same dignity and are coextensive. The only difference drawn by the Constitution is that only the natural born citizen is eligible to be President. Art. II, § 1.
While the rights of citizenship of the native born derive from Section 1 of the Fourteenth Amendment and ther rights of the naturalized citizen derive from satisfying, free of fraud, the requirements set by Congress, the latter, apart from the exception noted, “becomes a member of the society, possessing all t he rights of a native citizen, and standing, in the view of the constitution, on the footing of a native. The constitution does not authorize Congress to enlarge or abridge those rights. The simple power of the national legislature is to prescribe a uniform rule of naturalization, and the excercise of this power exhausts it so far as respects the individual.” —Schneider v. Rusk, 377 US 163 1964
Based upon the language of Article II, Section 1, Clause 4 and the guidance provided by [the Supreme Court of the United States in their 1898 decision in the case of U.S. v.] Wong Kim Ark, we conclude that persons born within the borders of the United States are natural born Citizens for Article II, Section 1 purposes, regardless of the citizenship of their parents. Just as a person born within the British dominions [was] a natural-born British subject at the time of the framing of the U.S. Constitution, so too were those born in the allegiance of the United States natural-born citizens.Indiana Court of Appeals, Ankeny et. al. v The Governor of Indiana, Mitch Daniels, Nov. 12, 2009
Any person which needs to resort to statute to prove his citizenship is naturalized.