In the short term, propaganda can get an entire country of people to do just about anything. In the long term, people come back to themselves and tell the charlatans to take a hike. The big danger, as borne out all across history, is to invest concentrated power in someone to take care of the problem. People like Cincinnatus are rare, and most people don't give up power once they have it.
The public discourse will eventually relegate the nutters. Lose Free Speech, not just in the letter of the law, but in the spirit throughout society, and you lose the public discourse. That's not somewhere we want to go.