Recently a paper was published that gives a proof of this statement with elementary methods. Among other things the authors also give a short discussion on your question. You can find it at https://arxiv.org/pdf/1707.08864.pdf

Recently, René Corbet and Michael Kerber posted the preprint "The Representation Theorem of Persistent Homology Revisited and Generalized" which gives a proof of this statement with elementary methods. Among other things, the authors also give a short discussion on your question.