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.
That's great but he wasn't born to an American citizen. According to your linked source that means he couldn't have been an citizen, which means his children and grandchildren couldn't be citizens either. Just following the logic of your reference.