I believe that citizen-at-birth is a necessary, but not a sufficient basis for finding "natural born citizen" for constitutional purposes, under the intention of the founders.
If one adopts the construction you advocate, then an infant born to illegal aliens, literally with allegiance to a foreign nation, and raised with allegiance to a foreign nation, is "natural born" and therefore eligible. But the writings of the day bothered to notice the allegiance of the parents. Your analysis doesn't notice the allegiance of the parents, at all.
Right or wrong (I think it’s wrong), the way the current law is written, the infant of illegal aliens, if that infant is born in the USA, is a natural born US citizen.
See paragraph “a” in post 128, which is a copy/paste of Title 8, Section 1401 US Code.
That’s why folks get upset over “anchor babies.”
Some lawyer can feel free to correct me, if I’m wrong.
Your belief is wrong. See US v. Kim Wong Ark.