A corollary of this seems to be that Harmonic analyses work fine when the notes do consistently line up according to their function, which happens all the time in pop music and possibly in Classical music although I'm not certain of this. Having said that, my biggest worry with Westergaardian theory is that it is almost too powerful. Whereas Harmonic theory constrains you to producing notes that do sound in some sense tonal

Note that when analyzing tonal music with Westergaardian analysis, it is generally the case that anticipation and delay tend to occur at relatively shallow levels in the piece's structure. The deeper you go, the more notes are going to be "aligned", just like they might be expected to be in a harmonic analysis. Moreover, the constraints of consonance and dissonance in aligned lines (as given by the rules of counterpoint; see Westergaard's chapters on species counterpoint) will also come into play, when it comes to these deeper levels. So it seems that Westergaardian analysis can do everything that you expect harmonic analysis to do, and of course even more. Instead of having "harmonic functions" and "chords", you have constraints that force you to have some kind of consonance in the background.