From my understanding, the Trump position, if effected, would not disqualify children of non-citizens from US citizenship if the parents are here legally, as were the Rubios. As far as I know (and, granted, that’s not really very far) we’re the only country in the world that grants citizenship to anyone and everyone born here, regardless of circumstances.
Yes, permanent non-citizen residents that was opined by Justice Gray in US v. Wong Kim Ark, 1898. Many believe that SCOTUS opinion is in error, but that's another subject.
Congress has the power to create naturalization, positive citizenship laws that is not in conflict with the 14th Amendment to cover anchor babies and the chain migration of alien subjects and citizens that follows.