"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens . . .."
This Amendment had a specific meaning, not a broad meaning. Its intent was to settle the citizenship of emancipated slaves who had hitherto been State-less, that is, in a kind of limbo with respect to jurisdiction because they had been property, not persons. The Amendment meant to state that once freed they became persons who came under the jurisdiction of the United States, whether they were born slaves here or abroad.
Indians were not understood to be subject to the jurisdiction of the United States at the time, but were subjects of the Indian nations, and were thus not granted citizenship by the Amendment in 1868, else the 1924 Indian Citizenship Act making them so had been unnecessary.
Children of diplomats born in the US while their parents were representing the countries whence they came were not thereby granted citizenship either, because they like their parents were subjects of foreign jurisdictions. Senator Jacob M. Howard of Michigan, the author of the citizenship clause, said the Amendment excluded “persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers.”
Thus children born here to illegal aliens, like their parents, are subject to the jurisdictions whence they came and should not be granted American citizenship now, unless it please the people to do so.
It's that simple.
And the Congress has every right to make that rule consistent AT ANY TIME if it has not been so in practice or in litigation, as Article One, Section Eight of the Constitution makes clear:
The Congress shall have Power . . . To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes;
To establish an uniform Rule of Naturalization, and uniform Laws on the subject of Bankruptcies throughout the United States;
Meanwhile the importance of jurisdiction for citizenship is otherwise seen to be paramount because no one except an idiot or a malcontent questions the citizenship of a John McCain or a Ted Cruz because they were born to citizens while living abroad.
And even if it could be proven that a Barack Obama was born abroad it wouldn't make any difference to his citizenship because his mother was a citizen of the United States and subject to the jurisdiction thereof.
Anyone in the world can renounce jurisdiction and pledge allegiance to another flag if one chooses, and it is entirely within our rights as Americans to set the conditions for welcoming as well as bidding farewell to those who do so.