I am not a philosopher except an armchair one, but I think the point of my this is considered postmodern is here

"In fact, the only sort of “progress” that the novel refutes is that human nature can be refined and improved, but this is a very conservative and religious notion. There is nothing postmodern about the idea that our vices keep us from realizing all the progress that logic and science can offer us. It is a decidedly traditional mindset that traces back to the ancient world and finds a more full expression in Christianity. It is quite clearly the vices of humanity that cause all of the misery in the novel. Adso comes close to drawing the attention of Bernardo Gui, the papal inquisitor, due to his erotic obsession with the young woman with whom he had sex (an act that he notes would not be worthy of reproach had he not taken vows). Many of the monks are overcome with unhealthy curiosity over forbidden books, leading ultimately to their deaths. Bernardo Gui is filled with hatred and anger, which leads him to condemn others to death. The abbot and Nicola are both obsessed with wealth and objects of art. And Guglielmo, the brilliant monk who is supposed to be sympathetic to the reader, is ultimately overcome by pride and anger, which leads to the final tragedy at the close of the novel (following his final encounter with the abbot, he says Eh no, questo Abbone non può permetterselo a nessun costo. Grazie frate Guglielmo, l’imperatore ha bisogno di voi, avete visto che bell’anello che ho, arrivederci. Ma ormai la sfida non è solo tra me e Abbone, è tra me e tutta la vicenda, io non esco da questa cinta prima di aver saputo, which mocks the abbot’s obsession with wealth and the abbot’s ring in particular, ending with the phrase “now the contest is not just between me and the abbot, but between me and everything happening here, and I will not leave this fortress before I know” ). In short, the message of Christ is ignored almost seriatim by an entire community of people who have devoted their lives to that message. And ultimately, what can we say, what does the narrator say? We are all human and we all fail."

The enlightenment and the modern world very much believed that you could improve human nature. The enlightenment especially (which is why many of them were not religious). Postmodernism finds this idea fallacious. It doesn't do so on religious grounds true and you are completely correct that it is something that the middle ages would agree with postmodernism on (but for very different reasons) but I've always thought that postmodernism came from the rediscovery of this fact after world war II as epitomized in Edvard Munch's the scream.