So statements like "We have yet to esteem the power of the lie." and "Even the physicist has his mythology. Your atoms, they are not real." are not about the lack of objective reality? His notion of the domination of the herd over thought is not about the fact that history is continual oppression of everybody else by a succession of hegemonic collectives? And 'The Genealogy of Morals' is not all about the inescapability of social control? The notion that one must "devalue and revalue all transcendent values" is not about radical skepticism to the point of moral confusion?
We see these themes in Neitsche because they are there. His aesthetic notion of morality and his aim at 'Perspectivism', where no viewpoint can be completely wrong, are pretty much modern post-truth.
But he is hostile to Christianity and its aftermath, and thus is not moved by the Left. So he was not as self-destructively caught up in rendering his entire worldview political and converting everyone. But that is not a real aspect of postmodernism anyway, it is just paranoid overreaction to its internal instability and our clinging to the products of our own culture unconsciously whenever we advance relativism.
Moving away from guidance by simple rules toward personal experience is postmodernism, morphing the anti-rule rules into a domineering dogma of guilt for ever having profited from artificial clarity, and pounding the table about it, is leftover Christianity. Recognizing the pervasive mechanisms of social control is postmodernism, seeing almost all social control as oppression and hating it beyond measure so that you feel obligated to trash privilege and worship victimology is leftover Christianity. Knowing that morality naturally eats itself is postmodernism, creating an internally inconsistent yet excessively ardent politics out of the idea is leftover Christianity. In each case, Nietzsche espouses the former and omits the latter.