The original argument against pushing extremists onto fringe sites was that the fringe sites were dark to law enforcement. I also don't buy that argument. It's a weak anti-encryption argument, and I also don't believe they're anywhere near as dark as LEOs claim. There might be some chatter that existed in the clear, but no one is currently plotting terrorist attacks out in the open on Facebook who will suddenly switch to Telegram if Facebook becomes party to civil suits.
Working back to your previous argument, the idea that the legal juggernauts of the big social networks will protect them while smaller sites self-censor is directly contradictory to what you say about fake news migrating to fringe sites. There's certainly less financial burden on a small site with less traffic to regulate what's posted, and as it stands, small sites do have to regulate what's posted, so it wouldn't be much more difficult. The only people who have broad exemptions over their liability for what's posted, currently, are the big social media sites. So let them spend that good money on their lawyers.
Opening up user-submitted content sites to civil damages would ensure that the largest social networks are the only ones that can afford to fight these cases in court. Smaller sites would become self-censored, even if they weren’t likely to be a target of a civil suit. This would lead to a further entrenchment of the largest social media sites, as they would go to bat for their users at least some of the time. They have to be seen to support their users, or else they would just find an alternative host that isn’t subject to the jurisdiction of the civil suits.
This whole idea seems like a nonstarter, impossible to implement, and with a laundry list of unintended consequences and is counterproductive to your stated goals of reduced propaganda. Instead of being primarily on centralized social media, fake news would be relegated to smaller fringe sites where it can’t be monitored as effectively as on the larger sites, further contributing to the echo chambers you argue against. Your intentions also seem antithetical to free speech between willing participants freely associating.