The interesting thing is, that the 14th amendment is the only place in the Constitution where citizenship is actually defined. It is defined as "all persons born or naturalized" which means that if the 14th Amendment is set aside then little Conservative babies born here won't be citizens, either.
The 14th Amendment gives no definition of citizenship. Perhaps you mean it's the only place that mentions people gaining citizenship, but that is not quite true either; Article I gives congress power "to establish a uniform Rule of Naturalization" (that is, of gaining citizenship after birth). Obviously, one can gain citizenship via naturalization independent of the 14th Amendment. The only thing the 14th Amendment does in regards to a
naturalized citizen is that it makes them an automatic citizen of the state they live in as well. There was previously more of a distinction between being a state citizen and being a United States citizen prior, which the 14th Amendment largely removed.
Still, I'm not sure what point you're trying to make here. No one has claimed that the 14th Amendment, or even just the Citizenship Clause, should be "set aside". Some people claim (incorrectly, in my view) that it doesn't give birthright citizenship to children of temporary or illegal immigrants, but that's not setting it aside entirely, but restricting it. There are some who admit it grants birthright citizenship to them and say it should be changed, but they're not saying to get rid of the thing entirely, just restrict it.
That said, if the entire Fourteenth Amendment were repealed, then... in regards to birthright citizenship, nothing would change, because the US later put the Citizenship Clause into the law itself, as seen in
8 U.S. Code § 1401(a):
The following shall be nationals and citizens of the United States at birth:
(a) a person born in the United States, and subject to the jurisdiction thereof;
It's redundant given the Fourteenth Amendment, but as it's part of an enumeration of categories of who gains citizenship at birth, it makes some sense to put it into law.
Or perhaps the claim would be that if
both of these were withdrawn, no one would gain citizenship at birth. Prior to the 14th Amendment, I think they more or less just went along with what the English common law said (which gave citizenship to people born in the US even if their parents were not citizens), but I'm not sure if it was actually codified at the time. But while a potentially interesting thought exercise, it isn't what anyone is asking for anyway.
