In Spivak's Calculus on Manifolds, he defines the integral of a $k$-form over a $k$-chain, and proves a version of Stokes' theorem for this situation, before moving on to discuss the integral of a differential form over a manifold. Other books, such as Introduction to Smooth Manifolds by Lee and Analysis on Manifolds by Munkres, seem to skip this step. They define the integral of a $k$-form over a $k$-manifold without mentioning or proving theorems about $k$-chains. (Am I correct about that?)
What is the advantage (if any) of Spivak's approach? To me, $k$-chains appear to be an unnecessary weird definition in a subject that already has too many unfamiliar definitions.