Ms. Coulter did not say that.
She did, she just handwaved it away and hoped you didn't notice.
Whether the children born to legal immigrants are citizens is controversial enough. But at least there's a Supreme Court decision claiming that they are -- U.S. v. Wong Kim Ark. That's "birthright citizenship."
US vs. Wong Kim Ark, 1898 held that:
The evident intention, and the necessary effect, of the submission of this case to the decision of the court upon the facts agreed by the parties were to present for determination the single question stated at the beginning of this opinion, namely, whether a child born in the United States, of parent of Chinese descent, who, at the time of his birth, are subjects of the Emperor of China, but have a permanent domicil and residence in the United States, and are there carrying on business, and are not employed in any diplomatic or official capacity under the Emperor of China, becomes at the time of his birth a citizen of the United States. For the reasons above stated, this court is of opinion that the question must be answered in the affirmative.
https://www.law.cornell.edu/supremecourt/text/169/649That is, on it's face, pretty clear. It's worth reading the decision, because it mentions in there that even though the framers of the Thirteenth and Fourteenth amendments obviously had freed slaves in mind (Ms. Coulter's argument) that those amendments apply to other races as well.
I agree that it's not cut and dried, as Mr. Wong's parents were in San Francisco legally, but that part I quoted above does not mention immigration status. It just says "permanent domicile", something that describes quite a few illegal immigrants.
There's probably room here for Congress to change the legislation and it to hit SCOTUS to for clarity, but pretending that the President can do it, legally, with an EO is a pipe dream.