Well, conservation of energy, which was, in its earliest forms, as mechanical energy or vis viva, applied to such problems, is a rather ad hoc method, whereas D'Alembert's principle is a more comprehensive solution. I understand that some teachers believe students are not savvy enough to appreciate such differences, but I think they are. In my own teaching, I would probably use both methods, thus showing how two different trains of reasoning can bring one to the same conclusion. But alas, I have not had occasion yet to teach anything about this problem yet. I just find it an interesting one, an interesting deficiency. A similar deficiency exists with the usual presentation of the pendulum problem, which likewise cannot be solved using Newton's method. Newton himself did a little bit of hand-waving in presenting his own solution.