No, only because we cannot come up with any HYPER specific definition of Hate speech. All the definitions are broad. "Words that inflict harm on people" ok, that could be EVERY single word in the English language, someone could be offended by puppies and believe that you are committing murder against them for saying the word "puppies". That doesn't make it hate speech. There is also context too, which is virtually impossible to define in black and white legal terms.
If we could create a hyper specific, succinct definition of hate speech where no one was confused or able to question the definition, then yes, hate speech should be legal. And yes, laws that do not have as strict definitional standards that I am proposing, should create those definitions.