I disagree. They certainly didn't destroy Karajan, or Thomas Mann, just two examples amongst many. Artistic culture continued to function very well... through "keeping ones head down" or "absconding". Who was better than Mann or Karajan in the stable democracies?
Karajan was a conductor... not a composer... not the creator of music, but an interpreter. Brilliant as he was, Bruno Walter and Toscanini are just two names that pop to mind of equals in the world of conductors. Thomas Mann left Germany and wrote two further books of real note: Doctor Faustus and The Confessions of Felix Krull. Literature seems to have survived the Nazi destruction of German culture the best. Mann and Hesse have a number of solid heirs: Grass, Boll, Paul Celan, Ingebourg Bachmann, Friederich Durrenmatt, Max Frisch, etc... but compare this to before the war: Kafka, Robert Walser, Rilke, Georg Trakl, Gottfried Benn, Thomas Mann, Hermann Hesse, Bertolt Brecht, Frank Wedekind, Joseph Roth, Walter Benjamin, Sigmund Freud, Hermann Broch, etc...
But let's look at music. Prior to the Nazis the Germans produced Mahler, Richard Strauss, Max Bruch, Joseph Rheinberger, Engelbert Humperdinck, Hans Pfitzner, Franz Schreker, Joseph Marx, Walter Braunfels, Egon Wellesz, Paul Hindemith, Karl Orff, Erich Korngold, Viktor Ullmann, Ernst Krenek, Kurt Weill, Othmar Schoeck, Arnold Schoenberg, Alban Berg, Alexander Zemlinsky, Franz Lehár, Emmerich Kálmán... How many Austrian or German composers can you name after WWII? Stockhausen? Henze?
As for who was better than Mann or Karajan in the stable democracies... have you been drinking? How many French, American, and British composers should I name? How many French, British, and American poets and novelists? How many of the above writers were blacklisted by Hitler and had their books publicly burned? Hesse and Mann fled to Switzerland. Trakl, Rilke, and Kafka died before the Nazis came into power. Walter Benjamin fled from France to Spain where he committed suicide when he found that he was to be turned over to the Nazi authorities in France. Freud finally fled to Britain. Kafka, Freud and others undoubtedly would have ended up victims of the Holocaust.
The visual arts suffered even worse in Austria and Germany. The early 20th century saw a Germanic Renaissance in art. Vienna and Berlin became major art centers. The Jungenstil/Sessesion in Austria, Die Brucke, Der Blaue Reiter, the New Objectivity, Dada, and the Bauhaus were among the major artistic movements of the first half of the century. Among the Austro-German artists were Gustav Klimt, Egon Schiele, Oscar Kokoschka, Adolf Loos, Otto Wagner, Franz Marc, E.L. Kirchner. Karl Schmidt-Rottluff, Erich Heckel, Max Pechstein, Emil Nolde, Otto Dix, George Grosz, Max Becmann, Mies van der Rohe, Oscar Schlemmer, Max Ernst, Kurt Schwitters, Paul Klee, Joseph Albers, etc... German art after the war was almost wholly irrelevant. Not until the mid-1980s did any German artists begin to make a mark on the international art scene... and only one or two have shown any real lasting power.

