Who’s driving cancel culture??

Totalitarian authorities don’t need a left or right...
we have had both religious (very conservative Roman Catholic style kings), patriotic (nazi style), left wing (Stalin style), right wing fascist (Mussolini) etc etc.

ALL of them oppressed free speech.

A free society built on liberty and freedom of the individual, the sanctity of person and property, will unfortunately, tolerate proponents of socialism. People can form communes and other forms of non-governmental groups such as religious sects, like the Amish or Mennonites. They might find it difficult because they have to control the individual and the general populace allows freedom of the individual. But that's another story. The point is that socialism can gain support in a free society, especially when people don't understand what it is. Reasoning and logic will keep it at bay for a time but eventually opposition weakens to the appeal of compassion and the necessity of government action in crises, such as floods, catastrophic events and poverty. Thus, the weakest point in a society that government can take over or invade is charity. It then appears to be a benevolent and benign force. It can provide public education, healthcare, protection from criminality and invasion from foreign forces. Onnce these things are achieved there is no taking them away. But then, what is provided as a "free" service to the public becomes greater in demand and taxes must rise to cover the increased demand. It's very difficult to argue that increased resources are necessary and generally they are given. At some point taxes cannot be raised further with out completely decimating the economy. The problem then is resolved by restricting services. Government by this time has reached a leviathan bureaucratic level and is now the master and not the servant of the people.
The crisis in Germany that allowed the rise of Hitler was the threat of communism. The holodomor had just occurred in the Ukraine and the fear of communism destroying Germany was palpable.
Hitler had been successful in driving out communists in some of the German city-states and he rode that wave to power. If Trump had done that he would have had people like Obama, Soros and Clinton arrested. But he was calling for reason and logic, law enforcement and the courts to prevail. Too many in power thought of him as an interloper though and he had little establishment support. As did Hitler, I suppose.
Hitler's manifesto reads similar to the communist manifesto, only on a national level. It was their similarities and not their differences that made them enemies, whereas, it is America's differences that makes it an enemy of socialism. Socialism pretends it is all about democracy and compassion until, and we see the shift now, we find it is about power, the centralization and grasp of it.
So yes, all forms of socialism are far left.