Those falsehoods comes mostly from old people watching fox news. Young people who grew up with mostly unfiltered communication on the internet are typically for doing something against climate change. So this only strengthens the point that free flow of information is more desirable than filtered communication.
People who want to manipulate the masses hates this though, so they try to make every argument they can that free flow of information is harmful and we need to stop it. Basically anyone with power have nothing to gain from you and me being able to freely communicate, so they want to reign that in as much as possible. I'm not sure why you want to make the rich and powerful even richer and even more powerful, but that is what you are arguing for.
It would be a horrible outcome if speech maximalism succeeded and prevented people from acting to protect their customers, property, society, etc., against destructive speech.
Of course, you can say "yeah but if you allow suppression of speech then it'll be used for evil" -- as in, Twitter can kick off whoever they want. But going as far as legally protecting any and all speech, in any situation and on any public or private property, is far more destructive in my opinion.
I think this is perhaps a straw man. Nobody is asking to protect unrestricted speech in all private venues.
Would it be so bad to make it illegal for large services that have private DMs to snoop/censor those, such as we do presently for large services that have private telephone calls?
The dichotomy isn't public/private. The phone company is private and they aren't allowed to tap your phone (unless they're doing it for the government), and they're not allowed to shut off your service because you called members of the "wrong" political party too often.
Your stance is super reasonable. But look around, many people in this conversation are effectively arguing in favor of totally, globally unrestricted speech, which I find extreme and untenable.
Not to mention the world is getting better only in a very narrow sense. Environmentally, we're entering a very dark period. The falsehoods surrounding climate change will cost us a great deal.