The best way to look at this is to consider who is excluded. Those born in the US who are NOT subject to its jurisdiction are a fairly precisely defined group. This group, according to common law, includes primarily foreign diplomats, essentially those with official diplomatic immunity, and armed foreign invaders.
Members of these groups are not subject to the laws of the US and thus children born to them on US soil are not citizens.
Everybody else, who are subject to US jurisdiction while on US soil, is eligible.
I seriously doubt the Congress and the States intended the legalization of anchor babies when they passed the amendment, but that is what it says.
Any other interpretation, IMO, requires ignoring the plain meaning of the words, similar to the way gun control people try to ignore the plain meaning of the 2nd Amendment.
I figure that the children of illegal aliens, if shipped out of the US with their parents, would not be subject to US laws.
The various Indian tribes, on their reservations, would also not be subject to the Jurisdiction of the US, though they could be if they moved off the reservation.