"Allowed" is the operative word. It didn't define it. There was still a process to follow.

The expatriation act of 1907 specifically called those children citizens and I don't think the Cable act is relevant here.

The Expat law of 1907 dealt with naturalization of children born abroad to citizens and that female US citizens would LOSE their citizenship if they married foreign nationals.

You had to file the paperwork at the local consulate. If you didn't the child would have citizenship and it was a mess. Lots of American servicemen, their war brides, and children dealt with this after WWII.