Excuse me, but you need to know about this, it is from the actual 14 Amendment of the USC:
Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws. [emphasis added]
Those entering this country illegally are not "subject to the jurisdiction thereof." This is most notable when they go to trial and cannot be sentenced to a punishment not allowable in their native homeland. Hence, those here as ambassadors and such that have children born in this country do not have children with US citizenship.
Our current problems with those here illegally go back to doctors and hospitals in California that found themselves with illegals giving birth and having no US identification. Without out such they could not issue a US birth certificate. Rather than let this stop them, they simply made up something and issued a certificate anyway. Now we have "US Citizen" children that are not really US citizens.
“subject to the jurisdiction thereof”
Ambassadors and their family have diplomatic immunity and therefore are not “subject to the jurisdiction thereof.”
Illegal aliens are, which is why they’re a high percentage of those in federal prisons.
You are mistaken when you say they cannot be sentenced to a punishment not allowed in their home country, by which I assume you are referring to capital punishment.
AFAIK, there is no legal prohibition against such punishments, though there may be treaty provisions with individual countries that apply. I believe the state department often asks states not to impose the death penalty for purposes of good foreign relations. But I don’t think there’s any way to actually prevent a state from doing so.
But the fact that such a person is being tried at all indicates he is subject to, etc.
In any case, the legality of the parents has no inherent bearing on the citizenship status of the child. Possibly it should, but at this point it doesn’t.