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“Well aren’t children born of aliens made, or declared, US citizens via naturalization legislation?”

No. Not when they are born in the USA.

The US Congress has the power of naturalization, and they can pass laws setting up the criteria for making someone a naturalized citizen. If they are born a citizen, then they do not need naturalization, since they are NBC/14th Amendment citizens.

“This section contemplates two sources of citizenship, and two sources only: birth and naturalization. The persons declared to be citizens are “all persons born or naturalized in the United States, and subject to the jurisdiction thereof.” The evident meaning of these last words is not merely subject in some respect or degree to the jurisdiction of the United States, but completely subject to their political jurisdiction and owing them direct and immediate allegiance. And the words relate to the time of birth in the one case, as they do to the time of naturalization in the other. Persons not thus subject to the jurisdiction of the United States at the time of birth cannot become so afterwards except by being naturalized, either individually, as by proceedings under the naturalization acts, or collectively, as by the force of a treaty by which foreign territory is acquired.”

http://supreme.justia.com/cases/federal/us/112/94/case.html

If they are born a citizen, then they do not need naturalization, since they are NBC/14th Amendment citizens.
So to you the 14th Amendment endows natural born citizen status to anyone born in the US, even the children of aliens, both legal and illegal?

Did you miss this right before your snippet?

The distinction between citizenship by birth and citizenship by naturalization is clearly marked in the provisions of the Constitution, by which
"No person, except a natural born citizen or a citizen of the United States at the time of the adoption of this Constitution shall be eligible to the office of President,"
and "The Congress shall have power to establish an uniform rule of naturalization." Constitution, Article II, Section 1; Article I, Section 8. By the Thirteenth Amendment of the Constitution, slavery was prohibited. The main object of the opening sentence of the Fourteenth Amendment was to settle the question, upon which there had been a difference of opinion throughout the country and in this Court, as to the citizenship of free negroes (Scott v. Sandford, 19 How. 393), and to put it beyond doubt that all persons, white or black, and whether formerly slaves or not, born or naturalized in the United States, and owing no allegiance to any alien power, should be citizens of the United States and of the state in which they reside. Slaughterhouse Cases, 16 Wall. 36, 83 U. S. 73; Strauder v. West Virginia, 100 U. S. 303, 100 U. S. 306.
Doesn't say they're all natural born citizens, just citizens. Not all citizens are natural born citizens.

And what of this from after your snippet? It's really important as it makes a critical distinction.

Indians born within the territorial limits of the United States, members of and owing immediate allegiance to one of the Indiana tribes (an alien though dependent power), although in a geographical sense born in the United States, are no more "born in the United States and subject to the jurisdiction thereof," within the meaning of the first section of the Fourteenth Amendment, than the children of subjects of any foreign government born within the domain of that government, or the children born within the United States of ambassadors or other public ministers of foreign nations.

You have to read this stuff in context as you well know.
Nothing makes these people natural born citizens, just citizens.
An alien visiting on a student visa retains his, or her, citizenship due to a dependent power...their home nation's claim to its own citizens.

Your snippet even emphasizes this...

The evident meaning of these last words is not merely subject in some respect or degree to the jurisdiction of the United States, but completely subject to their political jurisdiction and owing them direct and immediate allegiance.