You've used the present tense, and right off I can't think of any way the 14th acts currently to create NBCs. As for the past, I do think the intention was (among other things) to establish that freed slaves were NBCs. But this Bingham character we hear so much about said that the introductory clause was "simply declaratory," which implies that they were already something. I don't know exactly what they were considered to be in between the 13th and the 14th.
Do you believe their are any other criteria [such as parentage] that must be met, or is simply being born in the country sufficient?
It is sufficient.
My opinion:
No, the 14th Amendment does not "create" natural born citizens. Natural born citizens are those who are born citizens, and they are born citizens according to the ancient rule of citizenship that has applied for many centuries.
As you have noted, those who passed the 14th Amendment believed it was "simply declaratory" of the law as it already was. The 14th amendment created no new rule for citizenship.
The problem was that there were those who were committed to deny the operation of that historic rule to certain persons solely on the basis of their race and ethnic origin.
To be specific: Former slaves and other black people.
What the 14th Amendment did was to say, "Look. This is the rule. It's always been the rule. Race was never included as an inherent part of this rule, and it shouldn't be. You can't deny natural born citizenship to people just because they're black, and you can't deny natural born citizenship to people just because they are former slaves."
Now, we could get into a discussion of whether the freed slaves, having previously been legally considered property, were, after the Civil Rights Act and the 14th Amendment, "natural born citizens" or merely "citizens." I don't really know the answer to that question, and I don't think it matters. Those folks are all long gone, and none of those born slaves ever ran for US President. So the question of whether they, having been born property and then having that status removed from them, were theoretically eligible to the Presidency is an entirely theoretical one, and a moot one.
I can tell you this, though: Black persons born on US soil since the end of slavery and the institution of the 14th Amendment, except for the historical exceptions, were and are natural born citizens. And no, it doesn't matter whether their parents were US citizens at the time, or not.
And this basic interpretation is not just my interpretation. It's the official position of the United States.
The Supreme Court, in US v Wong Kim Ark (1898):
These considerations confirm the view, already expressed in this opinion, that the opening sentence of the Fourteenth [p688] Amendment is throughout affirmative and declaratory, intended to allay doubts and to settle controversies which had arisen, and not to impose any new restrictions upon citizenship.
In 1871, Mr. Fish, writing to Mr. Marsh, the American Minister to Italy, said: The Fourteenth Amendment to the Constitution declares that "all persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States." This is simply an affirmance of the common law of England and of this country so far as it asserts the status of citizenship to be fixed by the place of nativity, irrespective of parentage. The qualification, "and subject to the jurisdiction thereof" was probably intended to exclude the children of foreign ministers, and of other persons who may be within our territory with rights of extraterritoriality.
But what are "rights of extraterritoriality?"
Wikipedia has an article on the subject. I quote: "Extraterritoriality is the state of being exempt from the jurisdiction of local law, usually as the result of diplomatic negotiations."
So those who are exempt from the jurisdiction of local law are... foreign ministers, ambassadors, foreign royalty, and, in any real sense at all, invading armies.
Non-citizen immigrants do NOT enjoy any right of extraterritoriality at all. They are just as subject to local law, just as "subject to the jurisdiction of the United States," as citizens are.
And finally:
The foregoing considerations and authorities irresistibly lead us to these conclusions: the Fourteenth Amendment affirms the ancient and fundamental rule of citizenship by birth within the territory, in the allegiance and under the protection of the country, including all children here born of resident aliens, with the exceptions or qualifications (as old as the rule itself) of children of foreign sovereigns or their ministers, or born on foreign public ships, or of enemies within and during a hostile occupation of part of our territory, and with the single additional exception of children of members of the Indian tribes owing direct allegiance to their several tribes. The Amendment, in clear words and in manifest intent, includes the children born, within the territory of the United States, of all other persons, of whatever race or color, domiciled within the United States. Every citizen or subject of another country, while domiciled here, is within the allegiance and the protection, and consequently subject to the jurisdiction, of the United States. His allegiance to the United States is direct and immediate, and, although but local and temporary, continuing only so long as he remains within our territory, is yet, in the words of Lord Coke in Calvin's Case, 7 Rep. 6a, "strong enough to make a natural subject, for if he hath issue [i.e., a child] here, that issue is a natural-born subject;" and his child, as said by Mr. Binney in his essay before quoted, "if born in the country, is as much a citizen as the natural-born child of a citizen, and by operation of the same principle." It can hardly be denied that an alien is completely subject to the political jurisdiction of the country in which he resides...