It may not be any "easier" than Brown's proof, but the work of people like <a href="http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/">Francis Sergeraert</a> and others has created a nice conceptual framework for these types of questions.  See especially the paper, <a href="http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/AAECC-Effective-Homotopy-of-Fibrations.pdf">Effective homotopy of fibrations</a> by Romero and Sergeraert.  They have also implemented their algorithms in Kenzo.