But perhaps there simply is an inherent problem with being a platform for user-generated content and that it requires somewhat robust moderation to avoid being cancelled, sued, or arrested.
I think you need to define was "common-carrier" means.
As far as I'm concerned, once a message is deemed to be intended to be public, "common carrier" goes away for the service that is going to be broadcasting that message to others. The copper/fiber/etc. lines between me and Twitter are still a common carrier, because even though my Tweet is going to be public, the individual TCP packets going to Twitter are still intended to be private.
Any system for publicly posting messages that isn't moderated will be overwhelmed by bad actors, since bad actors will nearly always be kicked off of moderated platforms.