The problem is that we don't distinguish between necessary speech and unnecessary speech. Necessary speech is political speech (speech involving policies, programs, etc.--actual actions for the community as a whole to undertake), which in a democratic political process must have freedom.

All other speech is unnecessary (as far as the community as a whole is concerned). That means it can be curbed by the community as a whole through governmental action without harm to the democratic political process--though I personally don't think unnecessary speech should be curbed unless it is speech that causes actual harm (not just hurt).

It is O.K. for individuals to curb unnecessary speech. Unless the community as a whole is curbing speech through governmental action, it is only individuals' actions, and falls within the prerogatives of civic freedom--even if those individuals are the owners of speech-based enterprises (though in a democratic society those enterprises, too, can be regulated in any way the community chooses via the democratic political process.)

I see 'cancel culture' as an evolution of'political correctness'. It is a crude, heavy-handed attempt at forcing Liberal society to realize that the ethic of Liberal justice must be mutual respect (of a basic kind: taking one another into account). For that, "Equality Is All We Need" (here in Medium).