>None of these sentences make any sense if moral statements aren't actually propositional/truth-bearing.

This seems like a weak argument. What do you mean by "make sense"? For example, "Santa leaves presents under the tree" or even "Harry Potter attends Hogwarts" are also statements that make sense to a great majority of people - and if everyone were acquainted with knowledge of Santa or the Harry Potter universe, we could make the same statements you describe we make for moral claims too. Logical or grammatical sense does not get to the root of the issue which moral anti-realists are arguing is the actual problem. Of course you can make statements about what people commonly believe. However, you can also make statements against what people commonly believe; most moral discourse exists on denying one or more of the ways-of-speaking you claim.

The fact that moral claims are interpreted and parsed is not an argument against moral anti-realism, it seems like more of an argument ontly against moral non-cognitivism. It's entirely possible for a moral anti-realist to say either:

(a) Moral claims posit truth and falsity in terms of the fiction those claims pertain to - i.e. "Harry Potter attends Eton" is false.

(b) Moral claims aim to claim truth and falsity, but consistently fail to do so, due to lack of a strong enough epistemological grounding - i.e. "There are N planets in the universe", in which we don't have (and maybe may never have) access to the whole universe.

(c) Moral claims are neither true nor false, the epistemic grounding on which the arguments of truth or falsity is simply missing, and if it exists, its relation to other facts about the world is strange and strained.

(d) Moral claims are both true and false - perhaps this implies moral relativism, or a more generous form of (b).

The fact that we have, gramatically, accustomed ourselves to speaking as though moral claims are facts about the universe says nothing about the facts of the universe. In fact, this problem is well recognized by moral abolitionists as to what we should do, if moral fictionalism is not ideal.

By the logic layed out in your comment, we must also necessarily say that aesthetic realism is true, in which case "This food sucks" or "This painting is gaudy" are just as much valid claims as moral claims, and a relevant expert could shut down anyone who disagrees. Aesthetic realism is significantly less popular than moral realism. I don't have a link at the moment, but I read a paper a while ago which explained how attempts to distance moral realism from aesthetic realism are doomed to fail. They're a package deal. It turns out, vanilla ice cream is just bad, and if peope were equipped with knowledge of culinary aesthetics, they'd realize that chocolate ice cream is best.

>My main point is that it is really hard to be nihilist about moral facts without also being one about other sorts of facts as well.

This is the "Companions in Guilt" argument, but I think prominent moral anti-realist philosophers such as Richard Joyce have offered at least reasons to doubt that it's such a successful argument. That is, he has identified several reasons why moral claims are different from other epistemic claims. Nevertheless, he's also argued from another point of view - that is, the fact that one is unable to counter every single argument for a Christian god, does not mean that the Christian god exists. On the balance of arguments between atheism and a Christian god, for most peope who think about the problem, the case tends to fall on the side of atheism, despite the fact that there are rebuttals to certain aspects of atheism, and there are good arguments for the existence of a Christian god. When you consider the whole case for a Christian god, including many arguments, most philosophers fall on the side of atheism, due to the balance of the argument.