It may not be any "easier" than Brown's proof, but the work of people like Francis Sergeraert and others has created a nice conceptual framework for these types of questions. See especially the paper, Effective homotopy of fibrations by Romero and Sergeraert. They have also implemented their algorithms in Kenzo.
See also An algorithm computing homotopy groups by Pedro Real, which uses the ideas of effective homology to sketch an algorithm for the homotopy groups of spheres specifically.