Cancel culture is not responding to an argument with counter-argument. Cancel culture is responding to an argument with an effort to personally destroy whoever dared to voice that argument, exclude them from the discussion and silence them. That's why it is "cancel" culture and not "debate" culture - you're not supposed to criticize somebody, you're supposed to end ("cancel") their presence in the society. In older, harsher, times, this was done by literally murdering people, but we have, thankfully, evolved past that. Now it's done by removing these people from their jobs, blocking their social media accounts, preventing them (sometimes by violent action) from speaking on any public forum and attacking (again, sometimes violently) of anybody who dares to give them support and comfort. Much more civilized!
> In older, harsher, times, this was done by literally murdering people,
In past times, nonviolent forms of 'cancelling' that would seem familiar to us today were also employed. It was/is called 'shunning.' Our modern term is just a euphemism for that.
This looks like a semantic/identity disagreement. What you call cancel culture is the most extreme minority of cases of what I call cancel culture. It's common to serve one's biases by using a well heard label to refer only to the cases in some ideological context that are "doing it right", or, on the opposite end, to refer to the worst version of cases in that ideological context. Hopefully we're both trying not to fall toward the edges.