You say that Westergardian theory allows one to treat the music of Berg, Schoenberg, and Webern as belonging to the same school as earlier Western music, as if this were a point in favor of that theory. To me, it is a proof that the theory is both wrong and destructive, because my aesthetic sense says that music is crap.

This is not really true, for a variety of reasons:

1. Schenker and Westergaard do not claim that their theory can explain atonal music. A claim that Schenckerian/Westergaardian analysis helps explain tonal music is much stronger than the claim about atonal music, and should be evaluated on its own merits. In particular, we know that Schencker was aware of early atonal music, and didn't like it.
2. People's "aesthetic sense" seems to be quite dependent on their musical experience. Modern atonal music was the result of a very gradual development of taking existing (e.g. tonal) music and adding more and more "atonality" (whatever that means: some would say dissonance, others would talk about modulation, or complexity). People generally learn to appreciate atonal music by retracing these developments gradually, and listening to more and more challenging pieces. Thus, while your aesthetic sense says that this music sucks, this may not prove much.
3. There is plenty of music that was clearly "not written using harmonic theory" insofar as harmonic theory (e.g. as detailed by Rameau's Treatise on Harmony) postdates it. And yet, Renaissance and Baroque period music (and even a lot of secular Medieval music) is generally appreciated, just as much as music written after harmony-based theories became established.

If you showed that Westergardian theory gave a simpler explanation of the music that I like, that would help convince me that it was a superior theory.

I do agree that this would be quite relevant.