What is at the heart of the debate over Hart? It is this: both the classical and new natural law schools are wrong if they think that the natural law can be known, lived, and legislated in abstraction from tradition and culture, which is, at heart, theological. The classical view of metaphysics, at least as articulated by Edward Feser, presupposes an extrinsicist understanding of the relation of nature and grace, and reason and Faith, and is, therefore, not Thomistic. It’s as if Feser has not read, or just not digested, the work of John Milbank, Tracey Rowland, and Alasdair MacIntyre.
But, first, who in the natural law schools thinks "that the natural law can be known, lived, and legislated in abstraction from tradition and culture"? Natural law theory is in essence an argument for a practical logic; right back to the beginning it has pointed out that, like logic on the theoretical side, there is a sense in which the natural law is natural to us all (hence the word 'natural') and there is a sense in which it has to be developed out into its consequences, and that obviously requires (and is affected by) education and the like. This is explicitly pointed out by Aquinas; it's a common point among classical natural law theorists; and new natural law theorists, if anything, emphasize it even more. Consider the corresponding analogue with logic: people are wrong if they think that logic can be known, lived, and applied in abstraction from tradition and culture, for the obvious reason that the rational life can't be known, lived, and applied in abstraction from tradition and culture. Human beings don't spring out of the womb engaging in rational discourse proceeding from a pure natural power of understanding; we have by nature the capacity to engage in logical reasoning, but this has to be developed. Natural law theory just applies this same idea to practical reasoning.
This is why the appeal to Radical Orthodoxy and to MacIntyre doesn't work; none of these people are saying anything that is contrary to the basic idea in all natural law theory, because this tradition-based type of critique already presupposes that we share a human nature that allows us to participate rationally in traditions at all, and natural law theory, like logic, concerns what in human nature is required to participate rationally in the traditions in which we are raised. You can't accuse people of being extrinsicist, and therefore not accepting the Thomistic principle of nature being perfected by grace, if all they are doing is insisting that there is a nature to be perfected.
The same problem laces the entire argument. Kozinski says, "Of course, the error of the new natural law theorists is grave compared to such extrinsicism, namely, the adequacy of practical reason alone to ground and explain ethical theory and practice." But no one says that, whether classical or new. Every natural law theorist, regardless of preferred approach, recognizes that there are elements of practice that don't depend on "practical reason alone"; and if there were any who didn't, this would be a position entirely independent of their natural law account itself.
He then goes on to say:
In my view, both the classical and the new traditions neglect these four realities: 1) the mutually dependent relation of speculative and practical reason; 2) the subjectivity-shaping role of social practices; 3) the tradition-constituted-and-constitutive character of practical rationality; and 4) the indispensability of divine revelation in ethical inquiry and practice.
But classical natural law theory depends entirely on "the mutually dependent relation of speculative and practical reason": that's how classical natural law theorists argue for the first precepts of practical reason; both classical and new natural law theorists explicitly emphasize "the subjectivity-shaping role of social practices" and by extension "the tradition-constituted-and-constitutive character of practical rationality", because that's precisely how they explain disagreements in practical reasoning; and since the overwhelming majority of natural law theorists have been Christians of one stripe or another, they've never denied "the indispensability of divine revelation of ethical inquiry and practice" -- in fact, classically it's been common, going back to Aquinas himself, to point out that problems in our practical reasoning that are uncovered by natural law theory at least suggest a need for divine positive law.
This has been much of the difficulty in the discussion, that the critics in question repeatedly make their criticisms fly simply by attributing to natural law theorists positions that natural law theorists don't seem actually to accept, and never telling us what in the discussions of these theorists is supposed to back up these generalizations.
There are some interesting elements to Kozinski's article, particularly his appeal to MacIntyre's criticism of Maritain's "democratic charter". The problem, however, is that Maritain was entirely aware of the problem MacIntyre raises: it's explicitly noted by him on more than one occasion, he had to deal with it repeatedly in his rights advocacy (I like MacIntyre a lot, but Maritain, unlike MacIntyre, actually had to deal with the practical issues of rights advocacy from a Thomistic perspective on a regular basis, and so had a better understanding of them than MacIntyre's more academic standpoint really allows MacIntyre to have), and it is the foundation for his Communication with regard to the Draft World Declaration of Human Rights (PDF). Kozinski's use (against Feser) of MacIntyre's specific criticism of Maritain requires a conflation of natural law with natural law theory that would not be accepted by Maritain or Feser (or any other natural law theorist worth his or her salt): it confuses, that is, the account with what it accounts for, and thus a theory of reason with reason itself. You don't need a theory of reasoning, much less a correct theory of reasoning, for it to be true that you depend on the principles of reasoning for living any kind of human life. And, likewise, you don't need a theory of practical reasoning (like natural law theory), much less a correct one, for it to be true, if natural law theory is correct in its basic contentions, that living a human life depends on natural law itself.