That verbiage was what came out of a committee. The 14th amendment didn't look like that when it began, but because some of the senators were confused by this concept known as "local allegiance", they diddled with the original verbiage which was much more clear on the subject, and they put forth this current confusing mess in which "jurisdiction" is not clearly defined in context.
Senator Johnson understood the meaning to be this:
Now, all that this amendment provides is, that all persons born in the United States and not subject to some foreign Power for that, no doubt, is the meaning of the committee who have brought the matter before us, shall be considered as citizens of the United States.
By the way, that link above is to the debates on the 14th amendment. You can go forward or backward from there to read as much of it as you like.
I will also point out that Senator Jacob Howard explicitly said it will not apply to Indians because even if they are born within US Territory or a state, they are not subject to our jurisdiction.
Almost everyone coming up from South of the border is an "Indian" in the meaning used by Senator Jacob Howard.
Thanks.
Original intent is nice.