If you are going to assess how important myth is in Auden's poem "The Shield of Achilles," it is best to start with the most basic points, to establish a foundation.
Myth is incredibly important in this poem. In fact, it is not going too far to say that without myth, this poem would not exist. This poem is built around Greek mythology. To understand it, you need to be familiar with Greek mythology, especially as Homer uses it in his classic epic poem The Iliad.
In The Iliad, Hephaestus, god of crafts, the forge, and blacksmiths, makes a shield for Achilles. That shield was covered with images which summed up the Greek world in miniature form. It is as if Achilles was carrying Greece (or the whole world) with him into battle, and like it was protecting him.
Auden's poem describes another version of that shield, one that communicates a much darker world. To understand why Achilles' mother Thetis is so upset by what she sees on this shield, you need to know the mythological context.
Though it is a scandal to say so, in some ways, though, Auden's poem uses mythology in a more focused fashion than Homer. As his poem says, Achilles is about to die. It makes more sense for him to carry a shield marked by death and disappointment than by the rich life of Greece.
A final way Auden's poem uses myth is to comment on our world. If the original shield showed Greek myths of joy and life, Auden is here giving us modern myths of emptiness.