Even if climate change wasn't happening (it is), it seems plain to me we'd all benefit from less sulfer in the air. Not even coal miners like smog (though if they live far away from the smog they may be apathetic towards it.) The universal appreciation for fresh air should be more than enough motivation for anybody.
If your side is failing to convince people, the solution shouldn’t be “shut up the other side”, no matter how right you may be. It’s intellectually dishonest.
By engaging in cancel culture, it gives the silenced side way more credibility, because they can say “why are they afraid to let us speak?” It’s not just that cancel culture is morally bad, its also ineffective.
When the "other side" is arguing in bad faith, you're damned if you let them speak and damned if you don't. Someone like that isn't going to play by the rules of logical, rational discourse - such as sticking to the facts - so there isn't much to be gained by letting them speak versus not giving them a platform. I point you to Holocaust denialism, which has a long, rich tradition of ignoring, obfuscating, or concocting elaborate alternative explanations for any truths that run counter to their dogma. Or for something more benign, the flat earth movement or the Apollo landing conspiracy theorists.
It's easy for a bullshitter to make up more bullshit, and people love to believe "contrarian" bullshit so they can appear smarter than the next guy. Refuting bullshit takes time, energy, and effort that could be spent on more productive activities.
It's absolutely a worry that legitimate contrarian speech (I get that that's an oxymoron to some people) might be suppressed. But that's not what's currently happening.
An analogous historic example can be found in lead toxicity denial and denial of the harmful effects of tobacco. It took generations to chase away the bad-faith actors in the scientific and medical communities.
The payoff for anthropogenic climate change existing and us doing something is much higher than the price of it not existing and us doing something.