Citizenship by blood is not established in the Constitution but by the various modifications of the Immigration and Nationalization Act. In 1936, when McCain was born, the 1934 modified version of the Cable Act allowed transmission of citizenship via his mother as well as his father...
It wasn't until the Immigration and Nationality Act of 1952 that todays standards of overseas citizenship conference took shape. Citizenship in the US is a bit of a mess.
https://www.yalelawjournal.org/pdf/123.7.Collins_r35np7ug.pd...