I'm not so sure about the historical accuracy of the claim that representation theory was initially dismissed. They quote Burnside being skeptical of representation theory in 1897 in the introduction of his book Theory of Groups of Finite Order, but soon after he this he reformulated Frobenius' work on characters in a group/representation-theoretic framework. Burnside used representation-theoretic methods to prove his eponymous theorem in 1904, and representation theory appears in the second 1911 edition of this book.

A lot of people don't realize that the concepts of abstract groups, vector spaces, linear transformations, etc. were not particularly obvious to people, and a lot of results were developed in specific contexts that were later realized to be more general. After the development of abstract algebra, these subjects morphed into something much closer to the standard algebra curriculum we see today.