> the assumption behind this very essay is that there is such a thing as a rigid, singular concept of "truth"

I don't think that assumption is required by the essay. It says that people with orthodox privilege believe that anything outside their orthodoxy must be untrue. So it merely claims that certain people believe (often falsely) that they can conclude something is untrue because it's unorthodox. A crisp universal definition of truth isn't necessary for them to believe this.