Why scholarly papers focus on DAGs instead of DCGs (Directed Cyclic Graphs)
Let me mention one thing which is absent in the other answers: DAGs actually occur in applications. For example, a Boolean circuit is an ordered DAG whose vertices are labeled by logic gates. Dependencies of various sorts, like prerequisites for different parts of a project, also form a DAG. Wikipedia has many other examples.

Some problems can only be stated for DAGs. For example, topological ordering only makes sense for a DAG. Other problems can be solved faster on a DAG due to its restricted nature. This suggests studying algorithms on DAGs. For similar reasons, there is a lot of literature on algorithms for planar graphs – another naturally occurring class of graphs on which some algorithms run faster.