I wish this style of proof would penetrate through to postmodern philosophers. If everything that one person chooses to believe is true for them, and that person chooses to believe that there is an objective truth that precludes this statement, then you have a problem. This is not wordplay or an irrelevant point, any more than Godel's proof was. Either there are false statements about morality, or your claim that everything is subjectively true is also false. It's not a trivial point.
And I find that in practice, what gives is that postmodernists do in fact believe in absolute morality and it's all just a game to fool themselves and you into thinking otherwise. I see no reason to play that game any more, especially as I think most people would be repulsed if they took the time to work through what postmodern morality really is, instead of what it claims to be.
(This doesn't mean that we automatically know what the objective morality is, or whether there is one or many. It simply means that true subjective morality is fatally and irretrievably flawed. I say "true" because the moment you introduce an absolute, it is no longer subjective. Any absolute will do. Moralities can be quite complicated and nuanced, like utilitarianism, even as a utilitarianist must inevitably take their utility measurement as an absolute axiom. There isn't much "absolute" there, but it is non-zero.)
That said, reading the original article, I'm not sure that David Brooks would necessarily disagree with what I just wrote. I'm not sure he's advocating total relativism so much as observing that whatever concretes may exists seem to bear no particular relationship to the concretes that are claimed by anybody, religious (or aggressively non-religious), philosophical, or otherwise. I think there's a difference between merely observing and making claims about what should be true, and this op-ed strikes me more as an observation. So I guess I'm responding more to your comment than the article.