> I'll note that it sounds like your notion of free speech here actually seems to cover less speech than the 1st amendment, since you're explicitly excluding harassment and threats which the 1st amendment may cover. Of course, that's determining what is legal and illegal while free speech ideals are more abstract, so that's kind of an apples-and-oranges comparison)
Right, "free-speech" is a subset of "speech", and it aspires to optimize for the volume of free speech, which is to say that as long as people are free to threaten and harass, others will feel uncomfortable expressing their views. Of course, if those in power agree with you, the intolerant person will say "all is well", but the problems are that (1) sometimes we are wrong and (2) sometimes the person in power isn't going to agree with us and we find ourselves on the wrong end of our own speech-restricting precedents. Those of use who have lived in liberal democracies most of our lives may find it hard to believe that an illiberalism with which we disagree could ever come to rule, and so we take free speech for granted.
> You reject using Cancel Culture to silence those expressing abhorrent ideas. But what if someone isn't expressing a political view or ideology, but is instead performing bad actions? Like using racial slurs and insulting minorities, or shouting sexual harassment at women walking down the street?
As previously mentioned, harassment (sexual or targeted racial slurs) are already harassment. Note that "I think $race is inferior" isn't an abhorrent idea that falls within the realm of free speech, but "Get out of our neighborhood, you $slur" is harassment.
> As you mentioned, threats and harassment aren't protected under the banner of free speech ideals. And, depending on the circumstances, those actions may be protected under the 1st Amendment, so are not illegal. Do you object to using Cancel Culture against those who are performing threats and harassment?
Yes, I object. We should criticize the cancel culture movement; fighting cancel culture with cancel culture is self defeating. In general, I would advocate for legal protections that would neuter cancel culture (and provide a host of additional social benefits), such as decoupling health insurance from employment and strengthening employment protections. In many cases, harassment is already illegal, but proving harassment is difficult and this difficulty is exactly what cancel culture thrives on, so we need better tools--if an employer has to pay an expensive severance, they'll be less willing to cede to a mob. We could also prohibit them from terminating employees on the basis of their free speech altogether (provided of course that the "speech" in question isn't the employee representing the company in an official capacity). I'm not sure exactly which of these policies would be the most effective, but I'm confident that there's a lot of opportunity.
> As a follow up, what about abhorrent views being put into practice, such as the CEO of a company instituting racial segregation in their stores?
This is already illegal, and free speech ideals are perfectly consistent with the laws that condemn this behavior (instituting racial segregation isn't speech of any kind).
> If someone in a position of power publicly expresses abhorrent views, and under Refutation Culture we should voice an alternative viewpoint, can that take the form of a protest? If we march around that person's office holding picket signs refuting their viewpoint, would that be considered coercive?
I'm not sure what "Refutation Culture" is, but if you're talking about free speech ideals, then yes, it's perfectly fine to protest as long as your protest isn't harassing or threatening. There's nothing inherently coercive about expressing one's perspective; it's when there's a threat (either explicit or implicit) or when it's outright harassment (you're petitioning a politician in their workplace--this is healthy democratic behavior as long as the protests and messaging are peaceful in nature; petitioning someone's home or shouting death/rape threats constitute harassment IMO).