Studying language could simply be studying "the language." Yet we have a tradition of making literature the focus of the MLA. The idea might be that literature is the best language, that it provides exempla to be imitated. Now, being able to read literature does not mean that you can speak the language. On the other hand, a communicative approach with no literature does not mean that all students will learn to speak well either. Literature, even if it is not the "best," still carries prestige or honorific value, and increases vocabulary, and familiarity with basic vocabulary. You will see the 2,000 most common words over and over.
Nowadays, most young scholars are not writing on literature for their dissertations. There may be some novels involved, but only for their historical content. Poetry is pretty rare, now.
We still teach literature to undergraduates, but on a lesser scale. A few classes for the major. It's not a major that someone would go into because they like to read, per se.
Serious interest in linguistics is even more rare. Most people don't think that way about language.
So the odd bifurcation is that the field still thinks of itself in two, largely non-overlapping categories. There is literary study and teaching, and mostly historical research with a very minimal literary aspect to it.