If you took out that particular section of law, those persons would still be citizens under the constitution. Their citizenship doesn't depend on the statute. Cruz's citizenship depends on the statute.
Congress certainly has the power to define who is a citizen, that comes with the power to make rules for naturalization.
The SCOTUS cases cited above say that citizenship that depends on a statute is naturalized citizenship. That's what they say. You may disagree with that conclusion, but it's the conclusion of SCOTUS.
Incorrect.
We are a nation of laws. The government is controlled and bound by laws. The law defines those who are citizens at birth and do not need naturalization. Without the laws that define those who are citizens at birth, there would not be any naturally born citizens.