[N]atural law theory by its very nature purports to provide the general principles of the practical logic of human ends and means that among other things serves as the foundation of moral reasoning. This is why Hart is right that the two serious rivals to it are Kantianism and some version or other of the Nietzschean Will to Power: Kantianism is a position that in a sense also purports to give the practical logic serving as the foundation of moral reasoning, but it denies that any such logic can be a logic of human ends and means; and (the relevant sort of) Nietzscheanism holds that the foundation of moral reasoning is human ends and means, but these themselves have no practical logic. Hart's argument boils down to an attempt to force a choice between these two, but he appears to do so by begging the question: on the intellect side, he appeals to Kantian principles, and on the will side, he appeals to some kind of will-to-power approach. But this is just to assume that natural law theory is false in its account of practical reason (which is intellect insofar as it decides for will).
The whole issue of persuasion or convincing people is utterly irrelevant to the question unless one assumes will-to-power as the underlying basis of ethics. This point does not depend on anything to do with natural law theory; it's the whole point of the argument in Plato's Gorgias, and is part of the argument in the Republic, that if practical matters are to be tested by persuasion, this is equivalent to accepting that might makes right. (They both also argue that there is no logically coherent way to develop this view -- that any attempt to do it ends up contradicting itself.) Whether one accepts the Platonic argument or not, we need to be quite clear that the very coherence of any attempt to judge these matters by whether they will persuade nihilists or the masses or anything of the sort has often been called into question, and thus it should not be proposed as if it were an obviously unexceptionable neutral test. There are longstanding arguments that as a test it is quite exceptionable and entirely question-begging.
This also abstracts from more technical issues, like the question of whether Hart has quite characterized the Nietzschean correctly (he's vague enough that you could say maybe, but it's unclear how fair the result is to Nietzsche) or whether he has really appreciated how much Kant has to put into getting the categorical/hypothetical distinction in the first place. I'm willing to give him the benefit of the doubt on such matters, just as I'm willing to give him the benefit of the doubt and assume that I just don't understand the argument rather than attribute to him such a blatantly incoherent and question-begging one. But it was a very perplexing piece.