No, I don't see any contradiction. You are just making one up by ignoring the context.
Just because I use the word "murder" does not mean I am one.
And no, Freedom of speech does not include "hate speech" (putting aside definition problems). Why? Because the freedom of speech, like any other freedom, ends when you are infringing on other's rights.
"You are just making one up by ignoring the context."
Context no longer matters, Papa John was called a racist because he said someone else said the N word. You said something you don't think should be allowed to be said. That's contradicting yourself.
Freedom of speech only includes hate speech, does anyone ever object to love speech? What other speech needs to be protected?
Hate speech does not infringe on anyone else's rights because there is no right not to be offended is there?