I don’t think they made a mistake. If you read what I suggested you read, you’d see there is a difference in how that word was used in their day in a particular phrase — “subject to the jurisdiction.” They were this particular because they did not want the definition to be misunderstood. They were not setting up a redundant clause. They knew it could mean nothing else. So, to understand the passage, you need to understand the meaning of the phrase. They understood it. It is up to us to do the same.
I suggest you familiarize yourself with the meaning, origin, and history of the passage. A cursory understanding of a single word won’t help you comprehend what the phrase means.
I read the passages, again, in the U.S. Constitution, and the Declaration of Independence. They are crystal clear, and I could detect no penumbras or permutations. I guess I’m not SCOTUS material.
If you wish to argue from a spirit or intent of the law vs letter of the law, then I can agree with your position. Otherwise, words have meaning, and the meaning of jurisdiction and allegiance in the context used in the material I’ve already cited is clear.