You are subject to the jurisdiction thereof unless you enjoy diplomatic immunity
Offering your own conclusive speculation as to what a particular English language phrase might mean to your mind is interesting, but it takes us further away from points relevant to the question of whether Congress could by statute enforce the Fourteenth Amendment in a way that interprets "subject to the jurisdiction thereof" to exclude childen of foreign nationals here illegally and to what extent the Supreme Court would accede to that view.
It's not speculation, and it's not the meaning in "my mind". It's just the meaning. It's plain to anyone who spent more than two weeks in law school. In this instance, it's referring to territorial jurisdiction. This condition is absolute. You either do or do not fall within the territorial jurisdiction of the US.
Your reading the phrase only as it pertains to immigration. But, do you realize that same phrase was also included in the 18th Amendment - something that clearly had nothing to do with immigration or citizenship. That phrase is what gives the US sovereign power to prosecute people within its jurisdiction. If you're not "subject to the jurisdiction thereof", then you are beyond the reach of America's legal system. As such, you could commit murder, rape or any crime and avoid prosecution precisely because you wouldn't be under the jurisdiction of US law. Of course, such an exemption only applies to people who enjoy diplomatic immunity. Everyone else, is subject to the jurisdiction thereof.