This is true but incomplete. Anyone born in the US is a citizen except children of diplomats, hostile occupying armies, aliens in transit through US airspace or waters on their way between two non US destinations and prior to 1924, Indians subject to tribal Law. Everybody else is 'subject to the jurisdiction' of the United States, per the 14th Amendment and substantial SCOTUS rulings. The reason the mother citizenship qualification makes it true is that it precludes any of the listen exceptions.
True, I didn’t include any of the exceptions because they don’t remotely apply in this case.