You’re right. It’s also by international treaties, as well as the Constitution and fed law.
Problem is, the provisions are routinely ignored.
https://cis.org/Report/Birthright-Citizenship-Children-Foreign-Diplomats
You would be correct, but the children of illegal aliens are given citizenship as well as legal aliens.
NONE of them are natural born citizens as they are all born with the nationality of their parents, they very condition of divided loyalty that the founders put the natural born citizen clause in to prevent.