No, it doesn't. For the 14th Amendment they chose a phrase with a well established meaning, and this isn't in serious legal dispute. The phrase "under the jurisdiction" means people who are subject to the law. Illegal aliens are still subject to the law. They can be arrested like anyone else. Who isn't? Diplomats with immunity, and invading armies subject to the laws of war.
If you are born in the US, and not the child of a diplomat or invading soldier, you are a citizen. Just as you would have been had you been born in England.
"See here for a clarification on what the 14th was actually supposed to mean."
You can find web pages on the internet arguing for anything, including a flat earth. It's wrong. People start with the conclusion they want to be true and find an argument for it.
That's just it, I'm a citizen of New Mexico. Yes, I'm subject to Texas law, when I'm there, but I'm not a Texas citizen. Visitors are always subject to the laws of the jurisdictions they visit, but that doesn't make them citizens of such.
You are incorrect.