Before Hitler came to power laws against "hate speech" were already in place from the century before. He used the fact to depict himself as an outcast until he came to power and curbed freedom of speech to a far more significant degree.

Suffice to say that censorship was pretty prevalent through the history of Germany and it did not provide the advantages some had hoped.

And the allies did censor speech that is true.

> The representative of the Allied forces admitted that the order in principle was no different from the Nazi book burnings, although unlike the burnings, the measure was seen as a temporary part of the denazification program.

In that case the measure can be understood as the country was full of Nazi propaganda and alternative media was more or less dead. But it was indeed temporary and the Allies also wrote Germany's constitution, which explicitly forbid censorship.