If you think studying "culture" is something different from studying literature, that it leaves literature behind, I have a different perspective for you. While you may be studying rock songs or comic books, or the gendered construction of bicycle riding in the 1920s, you are still doing it from a literary point of view, because you are practicing a literary genre (the essay) to construct a culturally constructed narrative, a fiction. It is tautological to say that culture itself is culturally constructed, that the narratives we invent to explain our culture to ourselves (or someone else's culture to ourselves!) are essentially fictional or mythic, that they depend on certain archetypes, plot structures, and tropes.
By this I don't mean that these narratives are false. After all, novels are only "false" in the sense that they are about made-up people. The real fictionality of fiction is not that the people and situations therein don't really exist, but that it offers ideological constructs that are only one particular way of interpreting reality. Fiction that is true to life is true to our ideas about what life is, our own fictions, as it were.
It is easy to see that certain right-wing cultural narratives are false, or that they are right-wing. Logically, however, narratives of which we are approve share the same narrative structures, the same tropological structures, as those that we condemn.
So cultural studies is the construction of politically edifying myths of culture, and / or the debunking of "bad" myths. It leaves literature behind because it tries to go beyond certain literary narratives of culture (those that privilege the creation of works of literature like plays, poems, and novels), but it is simply a new set of aspirational narratives about culture. These narratives try to find value in cultural expressions not part of the old aspirational paradigm, but what is at stake remains the same.