I don't think you understand the distinction. Free speech, as a principle, exists to protect and allow all people to privately hold and voice any opinion without persecution from the government. This principle doesn't extend to people who are making speech on behalf of the state. That's not their personal opinions, that's the narratives the government wants to release to the public. It is their job to release this propaganda, and that's an action to fulfill the obligations of the work they've been tasked with. Keep in mind, public officials still obviously have freedom of speech as they're still people, however, this protection doesn't extend to what they do within the capacity of their offices. That's the big difference between someone like Kirk and someone like Streicher.
Also just to be clear, I'm not one of those free speech "absolutists", I specifically said that I think the exceptions that are currently defined federally for the 1st amendment are the golden standard. Btw these exceptions are obscenity, child pornography, defamation, fraud, incitement to imminent lawless action, true threats, fighting words, and speech integral to illegal conduct. These pretty much cover everything that needs to be an exception. They're clear enough to set objective standards, but also have some ambiguity to allow for nuance. I don't think hate speech covers anything that's not already covered, I also think that hate speech as a concept is inherently more subjective, arbitrary, and therefore more volatile than the already existing exceptions.