I am very confused by these claims, because MacIntyre was exactly opposed to relativism, and emotivism, which he saw as characteristic of the hollow shell of modern morality [0]. He acknowledges Nietzsche's criticism of the farce of Enlightenment morals. He appealed to telos as the objective basis for ethics and morality and in that book advocated for return to Aristote.
Perhaps you misunderstand what culture is. It isn't some kind of fiction we lay on top of reality that gets in the way of reality. It is a shared language of a people about reality and one that is not static, but hopefully developing, but at the very least changing. Science is itself a part of culture. You are born into a culture, which can be anything form pretty good to downright lousy, and the "dialogue" of this culture of a people with reality, and other cultures, moves the development of this culture.
Think of all the things you have learned in the scope of science. That aggregate of learning is culture. The presuppositions that science rests on is culture. This doesn't contradict the possibility of knowing the universal. Rather, it is through the cultural that you come to know the universal and through which you are better prepared to know it. We benefit from thousands of years of cultural dialogue. We cannot attain a very high understanding of reality without immersing ourselves in this dialogue of cultures spanning human history.
(Incidentally, as MacIntyre was a Catholic convert, one thing the Catholic Church makes possible is the existence of both the particularity of ethnos and the universality of the Church; "catholic" means "universal". A multiplicity of cultures sharing in the universal, avoiding both cultural parochialism and an alienated cosmopolitanism.)
[0] https://en.wikipedia.org/wiki/After_Virtue