“Persons born to US citizens abroad are United States citizens AT BIRTH, and before any documentation of the fact is ever issued.”
You appear to have missed that bit in the statute about the residency requirements needing to be met for the US citizen parent.
If Cruz’s mom had been age 18 at his birth (like Barry’s) instead of age 27 (IIRC), Cruz would NOT have been a statutory citizen at birth due to his mom failing the residency requirement of 5 years US residency after age 14 that was in effect at the time of his birth.
Pub. L. 82-414 § 301(b)
Any person who is a national and citizen of the United States at birth under paragraph (7) of subsection (a), shall lose his nationality and citizenship unless he shall come to the United States prior to attaining the age of twenty-three years and shall immediately following any such coming be continuously physically present in the United States for at least five years: Provided, That such physical presence follows the attainment of the age of fourteen years and precedes the age of twenty-eight years.