You may be a horse per the common law you describe, which is a part of “Positive Law”, but you are not a horse per “Natural Law”. And the term “natural born Citizen” is a Natural Law term. That is what the adjective “natural” means. By Natural Law, not Positive man-made laws, amendments, or treaties. Most people don’t know the difference between Positive Law and Natural Law. First year law students are taught the difference but as soon as they get involved politically they tend to immediately forget it, for political agenda reasons of their client or political party or personal aspirations.
“natural law” is (by adoption) a Catholic concept. We went through that quite thoroughly in high school. It has nothing to do, definitionally, or conceptually, with the “natural” in “natural born”.