"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?
Yes, I'm a citizen. My ancestors can be traced back to the colonies, before there was a United States, or to Ellis Island, and they were naturalized. My Great Grandfather, who came through Ellis Island, also followed the rules, he was employed, had a sponsor here, and obtained his citizenship prior to sponsoring and bringing over his wife and children. They were also naturalized.