Puritans left England to escape the English Cancel Culture. Upon arriving, most of them then implemented their own cancel culture. After Massachusetts cancelled him, Roger Williams founded Rhode Island on the belief that religious persecution by the majority was wrong (note: moral, religious, ideological -- whatever you call it, it's the same idea).
This idea of personal tolerance is itself a moral position. Republic government reflects the will of the governed with emphasis on the will of the minority as a FUNDAMENTAL moral belief of the majority. "Free speech only applies to the government" is untrue because that is simply an outgrowth of the governed. When the governed majority actually start believing that "freedom of speech doesn't mean freedom from results", you have opened Pandora's Box by shifting that fundamental moral belief.
Once you have decided Cancel Culture is moral, you are simply negotiating the terms.
The majority of the US believes homosexuality is morally wrong, but it's a personal decision/action. What if freedom to come out as gay doesn't mean freedom from consequences? What if the majority of people who believe abortion is immoral decide to lay on the consequences? There's a huge list of moral disagreements where the majority could decide to cancel the minority.
Finally, Cancel Culture ALWAYS leads to violence. If tolerance isn't part of the mix (and Cancel Culture literally means you aren't being tolerant), then either someone must change their fundamental morals or must start a conflict. Saying "I'm tolerant" and then persecuting everyone who doesn't say and do exactly as you think they should might be cognitive dissonance, but it isn't tolerance.