If that were true then the child of any foreign national who happened to be born on the base would be a U.S. citizen. And I don't think that's true.
“If that were true then the child of any foreign national who happened to be born on the base would be a U.S. citizen.”
It only applies to American military CONNECTED to that base. Even if the military lives off the base, they are still connected to that base and retain their American citizenship.