Tu quoque, right off the bat! And you are mistaken, i'm not begging the question, i'm pointing out that there are two different systems. To Americans, the English System of "natural" law is valueless, our very existence is a rejection of it. Regarding the English, our system has slowly encroached theirs, making ours the superior of the two.
So to answer your question, the American philosophy of natural law has great value to Americans. Indeed, it was the key to our freedom and independence. What do you suppose that was worth?
How can you appeal to natural law as if it pre-exists and underlies the existing law, how can you treat it as God's law, when it is contingent on the human laws of a given place and time?
Because it isn't. "Natural Law" as defined by a Monarch and his supporters is not the most objective source. It necessarily favors the status quo for those at the top of the social structure. It is subjective.
What the founders learned was objective natural law, something which remains true regardless of ones position in the social structure. They realized that individuals were basically equal and had inherent rights, not derived from a King. Among them the right of Expatriation, the Right to pursue happiness, Freedom from persecution for their beliefs, elimination of class barriers, and so on.
When these ideas were first realized, and when we realized that we should insist on them, every other government but one (Switzerland) was a Monarchy. Since then, Monarchy after Monarchy has fallen, to be replaced by systems more closely resembling ours. Today, Monarchy is an endangered form of government, and that is assuming if you are going to argue that it still exists at all.
As Aristotle's understanding gave way to Newton's, I would say that evolutionary forces had demonstrated our comprehension of "natural law" to be superior to that of the Old English version.