I propose that the excerpt below does not confer citizenship upon the children of illegal aliens.

[From Title 8, United States Code]

1401. Nationals and citizens of United States at birth
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; [my emphasis added]

(b) a person born in the United States to a member of an Indian, Eskimo, Aleutian, or other aboriginal tribe: Provided, That the granting of citizenship under this subsection shall not in any manner impair or otherwise affect the right of such person to tribal or other property;
 
Just think about it. Why is paragraph (b) necessary? Would not a member of an Indian, Eskimo, Aleutian, or aboriginal tribe already be a citizen of the United States if born here? Maybe not. Their tribes are recognized as independent  nations within the U.S. and not subject to the jurisdiction thereof. That means it is possible to be born within the United States and not be subject to the jurisdiction thereof.

I believe that an illegal alien or immigrant (parent) is not subject to the jurisdiction of the United States until (in the case of an immigrant) they take the oath of allegiance. It is that oath which puts them under the jurisdiction of the United States and in that oath they renounce their allegiance to any other nation, state, country, or authority. I submit that a newborn baby is under the jurisdiction of its parents, and its citizenship is that of its parents. That is precisely true of babies born to diplomats like Ambassadors of foreign nations.

If what I submit is not true, then the U.S. can take a newborn baby from its parents, then deport the parents and keep the baby. Not that we would, but it would be legal under the law if we interpret the law that way. Actually, that is exactly what happened with Elvira Arellano. She was deported but her son was allowed to stay. If the law had been properly applied her son would have accompanied her. His mother is a citizen of Mexico, owes allegiance to Mexico, and is subject to the jurisdiction of Mexico.

I'll add more to this after I check the provisions for passports for infants. I think that infants are documented on the parent's passport until they reach a certain age, then can receive their own passport. I also think that the U.S. can NOT put an infant on an alien parent's passport as a U.S. citizen, nor can they (we) issue a passport for the infant as a U.S. citizen. The alien parent would have to visit their own consulate to get their infant documented as a citizen of their own country. The reason I think that is that my oldest son has a birth certificate from the U.S. Department of State because he was born in Germany while I was stationed there on active duty in the U.S. Air Force. He does NOT have a German birth certificate!