"The amendment specifies that “all persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside.” If the framers simply intended to make citizens of any person born in U.S. territory, then that central clause has no purpose."
The Framers did
not intend "to make citizens of any person born in U.S. territory. They intended to exclude agents of foreign governments -- foreign diplomatic personnel. Foreign diplomatic personnel and their families have diplomatic immunity, and therefore "are not subject to the jurisdiction" of the United States. It's a clear carveout that has been applied in that exact fashion for over a hundred years.
I don't think anyone seriously wants to argue that illegal aliens are not subject to the jurisdiction of the United States, and therefore cannot be prosecuted for any crimes they commit here.
In terms of "intent", it has to be remembered that there were no such thing as illegal aliens when the 14th Amendment passed because we had no laws restricting immigration at that time. If you got here, no matter how you got here, you could stay. And that means that the Framers of the 14th could not possibly have intended to exclude the children of illegal aliens because the concept didn't even exist.
Maybe we should have amended the 14th once we started limiting who could immigrate, but we didn't. So, we're stuck with everyone except those with diplomatic immunity getting citizenship if they are born here. Not what I'd choose...but that's what it says.
It would have been extraordinarily easy to write an Amendment stating that all slaves freed by the 13th Amendment are hereby declared citizens, rather than the far more broad formulation that the 14th actually contains. So clearly, it has to apply to more than just former slaves or the wording makes no sense at all.