Nazism is a meme, just like stupid cat videos and the bible. It spreads from person to person seeking fertile ground in terms of an easy to identify scape-goat and an association with trouble experienced in a persons life. Allowing such memes to spread unchecked is playing roulette with open societies, the question then becomes whether you'd prefer free speech if it becomes the deciding factor in whether or not a thing like that can take root on a scale that it could cause a disaster or whether you forgo free speech to some extent in order to squelch the problem before it gets out of hand. Note that the Nazis themselves were not exactly free speech fans.
So even if it isn't a necessity it certainly will help to make it grow, and will allow it to grow faster.
Your rationale is exactly what every dictator throughout history has used to protect their power. Every, single, one. You either have free speech and allow all to express their ideas unfettered, or you will end up in a dictatorship sooner or later, guaranteed. To be clear, i'm talking about expressing ideas like "men and women differ biologically" and not speaking about threats/fighting words. Once you start punishing people for their ideas you're on the road to dictatorship, at least if every dictatorship in history is any guide.
Right, that's why all countries that do not have USA style free speech are dictatorships. Come on, that's not even trying. If anything the USA is more at risk of becoming a dictatorship than many other democracies.
You're not going to see any trouble anywhere - except on private property - for expressing ideas such as 'men and women differ biologically', note that no government you'd care to list here has ever suppressed speech like that.
But Nazism is on a different level, and if you're willing to go down that road protecting the free speech of Nazis you really have to be very optimistic about human nature. I keep hearing echoes of 'it can't happen here'. But I believe it can happen, and it probably can happen everywhere. The question is if we will let it and what it will take to stop it once the ball starts rolling.
You seem to defend the idea that 'limitations to free speech lead to dictatorships' by defining a dictatorship as 'any government that puts limitations on free speech'.
You could make just as a valid (and philosophers like Popper have) that allowing unfettered free speech has also led every society that tried it down the road to dictatorship.
Dictatorship is one of the easy to find local minima of human organization, that we have to fight everyday not to slip down towards. You do that by accepting the humanity of your opponents, finding reasonable compromises, and there's no reasonable compromise to be had with the Nazi ideology, just ask the dead.
So even if it isn't a necessity it certainly will help to make it grow, and will allow it to grow faster.