The Nazis that were hanged at Nuremberg trails weren’t killed because of speech or beliefs, they were killed because of their actions.
You are simply wrong in this case. We hanged Nazi propagandists, as we recognized that they were committing conspiracy to commit genocide.
https://en.wikipedia.org/wiki/Julius_Streicher
Most of the evidence against Streicher came from his numerous speeches and articles over the years.[72] In essence, prosecutors contended that Streicher's articles and speeches were so incendiary that he was an accessory to murder, and therefore as culpable as those who actually ordered the mass extermination of Jews. They further argued that he kept up his antisemitic propaganda even after he was aware that Jews were being slaughtered.[73]
Streicher was acquitted of crimes against peace, but found guilty of crimes against humanity, and sentenced to death on 1 October 1946.