[1] A presentation of class field theory in terms of group cohomology was carried out by Claude Chevalley, Emil Artin and others, mainly in the 1940s.
The first reason that can be cited for that is that it did not provide fresh information on the splitting of prime ideals in a Galois extension; a common way to explain the objective of a non-abelian class field theory is that it should provide a more explicit way to express such patterns of splitting.
[2] The cohomological approach therefore was of limited use in even formulating non-abelian class field theory.
Behind the history was the wish of Chevalley to write proofs for class field theory without using Dirichlet series: in other words to eliminate L-functions.
[4] As of the early twenty-first century, this is the formulation of the notion of non-abelian class field theory that has widest expert acceptance.