Do only moral realists think ethics "is a thing"? No.
Consider this recent remark from J. P. Andrew:
Moral antirealism doesn't hold that ethics isn't "a thing," nor does antirealism prohibit one from constructing an ethical theory. It may not be able to provide one that is "correct" in the sense realists want a correct theory, but it can provide, among other things, useful theories. And I'll take useful over "correct."
Compare this remark to a remark one might make about the culinary arts. A gastronomic realist is someone who believes there are stance-independent facts about which foods are good or bad, and which foods you ought to eat or not eat. As a result, it may be the case that you ought to eat certain foods even if you don't like the, or you may have an obligation to prepare food in a way even if doing so would be contrary to your preferences. For instance, the following gastronomic facts could be true:
You should eat chocolate ice cream instead of vanilla.
One should never pair a white wine with red meat.
If given the choice, you may never choose taco bell over a restaurant with a Michelin star.
Imagine a gastronomic realist who said the following:
In my view, too much attention has been given to ethical anti-realism in the metagastronomic literature and the field of culinary arts. I’d rather see those who think gastronomy is a thing spend their time attempting to construct the correct gastronomic theory than argue with nihilists and relativists.
In other words, a gastronomic realist could maintain that only gastronomic realists should participate in discussions about the culinary arts. After all, those of us who don't believe that there are stance-independent gastronomic facts don't think gastronomy is "a thing" and thus we don't have anything to discuss. Only those who think there is a correct account of which food is good or bad and what our gastronomic duties are should proceed with building the "correct" theory of people should and shouldn't eat.
This would be completely absurd. You don't have to endorse gastronomic realism to think the culinary arts "are a thing," or to theorize about and discuss new ways of preparing food and drink.
It is no less absurd in the case of ethics. Those of us who deny moral realism aren't denying ethics "is a thing," in any and all respects; we could simply deny that the realist's conception of what the project of ethics consists in is correct. And moral realists don't own normative ethics. A search for the stance-independently correct normative moral theory isn't the only task to which someone interested in normative ethics can pursue, any more than attempting to identify the stance-independently "correct" foods to eat is the only worthwhile task in the culinary arts.
Quite the contrary: the pursuit of a "correct" account of which foods are good or bad in a realist sense is a ridiculous waste of time. So, too, is the pursuit of a stance-independently correct account of the moral facts. There is no such thing. And normative ethics should not be the exclusive domain of a futile dispute for which resolution is no more possible than a dispute about whether one has a duty to abstain from pineapple on pizza.
Note, too, that attitudes like J. P. Andrew's can contribute to precisely those self-selection effects I discuss when addressing philosophical expertise and other metaphilosophical concerns with academic philosophy. To the extent that those working in the field are sympathetic to Andrew's views, this can have several effects:
(1) Creating a literature that carries implicit presuppositions, e.g., realism, such that those who don't endorse those presuppositions have little way to enter into and engage in substantive normative discussion. This can make entry into the field difficult regardless of whether philosophers in the field would be receptive and welcome of those with contrary views (i.e., antirealists).
For comparison, if all practicing chefs agreed cheese was disgusting and never used in their recipes, someone inclined to use cheese in their recipes would encounter a field where their view simply wasn't reflected in the literature. This might discourage them from working in that area.
(2) Cultivating this attitude can lead to a kind of orthodoxy that can be socially enforced as well. If antirealists try to enter the field, publish, or discuss topics without sharing the realist's presuppositions, this could prompt those within the field to discourage them or even express outright hostility.
Both the content of the field and the social norms and attitudes within the field can work in tandem to create an orthodoxy with self-protective features. The result could, in turn, result in a stagnation in the field as its proponents no longer find their presuppositions subject to sufficient external challenge.
So should we leave normative ethics to moral realists?
No.
Note that Andrew is not saying that we should actively exclude moral antirealists; only that Andrew would prefer to engage in the latter question rather than constantly engage with antirealists. Nevertheless, such sentiments, expressed often enough, can serve to subtly reinforce the negative effects outlined above.
I'll end by emphasizing a general objection I have to the way philosophers frame issues exemplified in Andrew's remarks. Andrew's remark indicates that only moral realists think ethics "is a thing," which implies antirealists don't think ethics "is a thing." Some antirealists may not think ethics "is a thing," but the problem with this remark is that it is ambiguous and, more importantly, presupposes that only a realist's conception properly regards it as "a thing."
Yet an antirealist is free not only to deny moral realism, but also deny that the realist's conception of ethics is the only possible (or at least only correct) conception of ethics that regards it as "a thing." I often see philosophers frame the dialectic in such a way that seems to presuppose certain elements of their position are correct.
If antirealists are correct, it may very well be that the antirealist's conception of ethics captures it as it actually is, whereas the moral realist's conception doesn't. In which case, it is actually think moral realist who (on the antirealist's conception) fails to appreciate ethics as it actually is, while mistakenly endorsing a false conception of what ethics is. I raised similar concerns here.