Remarking just to point out that the analysis proceeds from a negative, "not owing allegiance to any foreign sovereignty." This is counterintuitive to the way people think. People think in terms of what they (or their parents) HAVE, not it terms of what haven't got.
Many things "shake out" in a way that most people find agreeable and comfortable, when the framework described by Bingham is applied. It takes care of anchor baby "natural born citizens," citizenship of children born to US citizens abroad (although I think the existence of the 1790 Act ironically makes that case HARDER, not easier).
Trying to state the principle in positive terms (what somebody has) instead of negative terms (what they don't have) results in some fairly interesting rhetorical gymnastics.
I think many people are uneasy about this because it would suggest that we are not all equal and that is antithetical to what we have all been taught about America.
To them I would say, chances are, you weren't going to be POTUS anyways so relax and abide by the constitution as our founding fathers wrote it.