Is there any reason to believe that if free speech wasn’t upheld for nazis something about the material history of fascism and its rise in the Western world would’ve been reversed? The hard left in the West has always been pretty marginal and every means, legal and otherwise, has been used to suppress them; this could lead you to the conclusion that discarding free speech would have no effect on your ability to organize but it would hurt the fascists, but it’s currently an uphill battle for the state to dismantle left wing media organizations.

No free speech protections would make the process of suppressing left wing dissent a lot faster. The government could directly take down anti-imperialist media for promoting terrorism, when currently such media can exist but at a disadvantage in the social media algorithms.

Your observation would hold if the states in the capitalist world had a stronger interest in suppressing fascism than communism. Maybe in that world, some communists would get arrested for praising China on the internet, but far more racists would be getting arrested and re-educated and somehow that would be a positive trade-off. The opposite is true, they’ll use any additional power to hurt communists before fascists, so it’s generally better for the state’s mechanisms of repression to be weaker and anyone who can agitate to that end is probably doing you a favor.

As for anti-authoritarianism in general it’s been a very successful psyop to make western leftists forget that serious leftist criticism of the state has always held that all states are authoritarian.