There were many people like Webern who closed their eyes for unwelcome realities, and being some 'other-worldly' artist, surely helped. As we know, in circumstances where surrounding realities are worrying and confusing, people simply compartimentalize their experiences, so it is no surprise to see someone who believes in nazi ideology helping Jewish friends, as Webern is said to have done. More interesting is Webern's idea of structure, which is thoroughly rationalistic and his approach of the 12-tone idea a top-down hierarchical organisational one, while every musician knows that there are many different, and more effective, ways of organizing 'unity' in music (as the repertoire shows). Webern's intention to make his music expressive is shown in his instructions where minute details get annotations like 'schmachtend', but these quasi-expressive moments cannot breath within the prison of a totalitarian system. And here lies the connection with nazism: an attempt to organize material in a grid-like way, entirely ignoring the properties of the material. Webern must have thought of nazism as a kind of 'clean' engineering process, until only the Best and Pure would remain, as he tried to achieve in his later works. With the messiness of life and the reality of people he had, obviously, quite some problems, which is already shown in his preference for the high mountain ranges of nature where vegetation has disappeared and only stone, snow and the cold wind of purity reigns.
Nothing demonstrates Webern's misconceived idea of 'unity in music' better than his example of the 'magic square', as he presented in a lecture in the thirties:
S A T O R
A R E P O
T E N E T
O P E R A
R O T A S
The letters read in any direction produce the same result. Webern used this example to show what Schoenberg, Berg and he tried to do with the 12-tone system: to create unity, in the same way as the greatest composers in the past created unity through thorough organization. But those composers used tonality, the basis of music, to organize their music, and their strict counterpoint (because that was where Webern was talking about) is only one way of organizing unity in music. Webern's example is a ridiculous self-defeating one, since the 'message' conveyed by the magic square is, in fact: 'The sower Arepo keeps the work running'. Who is helped by this communication, and how? The result of thorough organization producing nonsense.
As for Berg: I think he was too musically-aware that he had to preserve a more traditional, tonal, musical outlook, and he clearly never 'needed' the 12-tone idea; Wozzeck sounds verged at the brink of insanity and chaos precisely as Lulu, but only the latter work does use the system (which he found terribly cumbersome - why use it at all, one thinks?). His violin concerto is a heroic attempt to combine musical energies with anti-musical rationalism, an impressive feat and, it seems to me, entirely pointless: that piece could perfectly do without the system which cannot be perceived anyway in performance. It is like showing that one can do two complex things at once while one of them is pointless in itself, let alone in combination.