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.