Actually, it excluded aliens, as well. By saying they were subject to the jurisdiction of that state, what it meant was they were no longer subject to any foreign jurisdiction, an issue with illegal aliens
See here for a clarification on what the 14th was actually supposed to mean.
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.
"In other words, only children born to American citizens can be considered citizens of the United States since only a American citizen could enjoy the "extent and quality" of jurisdiction of an American citizen now.
That means that in all probability you aren't a citizen. Did your ancestors on both sides obtain citizenship before they had their children? If not, all descendants of those children could never be citizens unless they went through the naturalization process. How many US-born people do you think have gone through naturalization?