One line of argument often employed against Moral Realism appeals to Moore's distinction between 'natural' and 'non-natural'. With an adequate* lack of clarity, it can be represented as follows.
(NN1) Moral entities, if they existed, would be 'non-natural'. (Non-Naturalism)
(NN2) There are no 'non-natural' entities.
(NNC) There are no moral entities.
(Of course, Moore himself did not argue that moral entities do not exist—he defended a realist position.)
One crucial reason why 'the' argument from Non-Naturalism is not (cannot be) particularly significant is that it is unclear what 'this' argument could actually be. One crucial reason for this, in turn, is that it is quite unclear what the term 'non-natural' is actually meant to aim at. Notice first that the term has no particular customary meaning: it is used only within philosophical debates. Now, furthermore, within philosophical debates, the term is used in a fair number of quite different ways. Ironically, the secondary discussion about what the term 'non-natural' may actually be used to pick out has become more vivid than any primary issues, such as the truth or falsity of Moral Realism. As the SEP puts it: "There may be as much philosophical controversy about how to distinguish naturalism from non-naturalism as there is about which view is correct" (Ridge, Michael, "Moral Non-Naturalism", The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2018/entries/moral-non-naturalism/>). Given that the term 'non-natural' is unclear, the argument from Non-Naturalism is unclear; and since this argument is unclear, it surely makes not much sense to ascribe any privileged significance to it.
Another crucial reason against any privileged significance of 'the' argument is that, as a matter of fact, its prominence is derived from its heritage: it is supposed to go back to one of the chief arguments in one of the chief books (Principia Ethica, 1903) by one of the chief philosophers of morals (G.E. Moore). Yet as a matter of fact, Moore never stated 'this' argument. Contrary to widespread misunderstanding (and despite misleading wordings in the statement of the argument) in the relevant argument Moore actually did not invoke the 'non-natural' nature of moral entities. What Moore aims at is not that moral entities, if they existed, would be 'non-natural'; his argumehnt, instead, is that the notion 'good' cannot be reduced (and hence, as he thinks, cannot be subject to definitional analysis). The view that 'Non-Naturalism' goes back to Moore at all largely rests on an error.
Thus, G.E. Moore never stated an argument to the effect that moral entities, if they existened, would be 'non-natural' (as opposed to irreducable), and the notion 'non-natural', on which everything in 'the' argument hinges, is quite unclear. Having thus no actual connection to the admired great philosopher, and being in fact too unclear to be even identifiable, 'the' argument from 'the' 'non-natural' nature of moral entities certainly cannot be counted as an argument of any privileged significance.
* The lack of clarity involved, particularly in (NN1 and NN"), is adequate because it mirrors a lack of clarity in the argument itself.