A little rusty but not crusty
Well, in US v Ark (1898) the question presented to the court was "Is a child who was born in the United States to Chinese-citizen parents who are lawful permanent residents of the United States a U.S. citizen under the Citizenship Clause of the Fourteenth Amendment?" The key being "lawful permanent residents" and made no ruling for those that are here illegally. And it is and has been argued that that court disregarded the legislative intent with regard to the "and subject to the jurisdiction thereof" when crafting their ruling. A key participant in the framing of the 14th amendment was Sen. Lyman Trumball and according to his accounting of that phrase as it was being debated, included "not owing allegiance to any other country" and not just being present on US soil when the child was born.This just shows how little Trump understands the law and the Constitution. His proposed Executive Order would have as much meaning as a couple of pieces of toilet paper. Birthright citizenship is part of the Constitution (14th) and was codified by the 1898 Supreme Court decision. The sad thing is that there are some out there who erroneously believe that Trump would have that power just because Trump said so. Supposedly, Trump is arguing that children of foreign nationals born in the U.S. are not subject to the jurisdiction of the U.S. which is completely false. The only case where that would be true would be in the case of children born to foreign diplomats who enjoy diplomatic immunity and thus are not subject to the jurisdiction of the U.S.
You are correct in that the newborn’s parents could be given a choice of leaving the U.S. citizen baby in the U.S. while they return to their home country or takin the baby with them.