First of all, yes, private entities absolutely can ban you from speaking on their property. The 1A only restricts the government from jailing you. If we couldn't ban people from this sub, it would be an unreadable mess.
Second, the problem with the lefts cancelling is it was usually just a handful of terminally online freaks, digging through hundreds of tweets dating back years to find one example of a person saying "nigger", usually as a joke. It was completely devoid of nuance and purely vindictive.
The right is using it to get women who work with children fired because they're glad a normie was murdered. Does any normal person want a sociopath like that near thier kids?
The problem is always just the left, not the tools they use.
The left has determined that negativity is evil. They're dirty hypocrites about it, but their ideology definitely holds that "Inflicting pain or suffering of any kind is inherently evil (Unless you're doing it to someone inherently evil, in which case it's totally justified.)"
Ironically, I could find that core stance to be somewhat respectable, except that they have no concept of nuance for either end of the spectrum. So they deny pain where it's necessary; like natural consequences, character building, or corrective discipline, and embrace it whenever it suits their bloodlust to "righteously" destroy "evil."