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.
If everyone agreed that politics had to be kept out of work, then we'd be fine. But instead it's that your company is going to use HR to push Leftist indoctrination on you, while using profits to fund Leftist NGO's to literally harass and threaten you because you have anti-Leftist opinions.
If we could separate work and politics, than we'd be fine. But this was categorically denied to us, and will not be stopped until HR starts forcing people to watch John Birch Society videos for annual de-sensitivity training.