I agree that whether "the standard set of harmony concepts" is actually superseded by Schenkerian/Westergaardian analysis is not really obvious.

If you don't find it obvious after studying Westergaard and comparing it to (say) Piston, then my best guess is that you're relying on tacit musical knowledge that you don't realize others lack, or which you mistakenly think is being communicated in Piston (etc.) but which actually isn't.

Westergaard has a highly non-trivial theory of what counts as "consonance" or "dissonance" in a melodic line, which is roughly equivalent to "harmony" in standard music theory.

Not so -- there is nothing in Westergaard about root progressions (Rameau's "fundamental bass"), which is the defining concept of "harmony" in the traditional (theoretical) sense. Consonance and dissonance are part of traditional contrapuntal theory, which goes back to long before Rameau. (Yes, Westergaard does draw on the tradition of contrapuntal theory, as did Schenker.)

The other way that traditional "harmony" is recovered is that this kind of analysis allows for a note in the 'background'/'deep' structure to be tonicized over, effectively becoming a "temporary tonic" and admitting the construction of tonic triads ('arpeggiation').

Again, if you think this is what is meant by "harmony", you are missing the point. (Yes, Rameau kinda sorta had this idea as part of his theory -- but not really. It's really a Schenkerian idea.)

In harmonic theory, the "hierarchy" has only two levels of structure: a note is either part of the chord, or not part of the chord ("nonharmonic tones"). In Westergaardian theory (as in Schenkerian theory), there is no limit to the number of levels. Take the Mozart analysis that folds out from the back of the Westergaard book. The data in that analysis cannot be expressed in terms of harmonic theory. The latter is simply not rich enough. All you can do in harmonic theory is write Roman numerals under the score, which (at best) might be considered roughly equivalent to showing one level of reduction in the Westergaardian analysis (though not really, because the Roman numerals only contain pitch-class information, not pitch information like the Westergaardian version; plus harmonic theory's "chords" frequently and typically mix up different levels of Westergaardian structure).