What is the material difference between a web site and a BBS?
BBS operators would be equally responsible, and won't run BBSes. Same with chat rooms and IRC channels — their operators could be sued, so they won't run anything like that for public service.
Of course, a web forum could still be run over Tor, hiding the webmaster and the participants. There would be no one to sue, so whatever inflammatory content were on such a forum, it would stay unless the moderators cared.
If the point is to stifle the public discussion and push it underground, then removing the protection of operators from liability for UGC is the way to go.
A BBS in the days of yore didn't have deep pockets for anyone to go after. They did however get sued or taken down for copyright issues and other content. There's the infamous case of the Secret Service raiding the offices of Steve Jackson Games [0] over content that was in an upcoming book for the GURPS role playing game [1].
Open source software has been massively buoyed by the availability of platforms like Github and before them SourceForge. Before these platforms OSS maintainers had to manage their own version control, packaging/distribution, and issue tracking. Without those things in place (managed or self hosted) it's pretty hard to collaborate with others. Big projects could afford to do it (both money and effort) but most hobbyist projects would just dump a tarball on their university or ISP provided web space.
Services doing that laborious work for no cost has let OSS authors more easily collaborate and just get work done on projects. They've also enabled small projects to just exist since the author doesn't need to even know about the infrastructure needed to host their code.
You can, and many do, self host Github equivalents. It's not like Github used all the oxygen in the room and monopolized source control.
The same is true for web fora. You can go host your own forum/fediverse site right now pretty cheaply. Domain names are cheap and TLS certs are free. There's nothing stopping you or anyone else from doing that. Plenty of people already are doing so.
Centralized platforms come into being because of network effects. You can host your own forum (or whatever) but that doesn't mean people will come join it. Lots of groups formerly served by on-topic forums moved to Facebook groups because all the participants were already there. It's no-cost vs low-cost and all of the infrastructure is managed by Someone Else. Infrastructure maintenance is a pretty thankless task.
Starting a new group on Facebook (or wherever) is pretty frictionless if all the participants are already on Facebook. There's a lot more friction starting a new little island of discussion with a forum.
By wanting to go back to the "BBS days" you're wanting network effects to not be a thing that exist. You're also somehow expecting people to have the technical chops to run a site. In the "BBS days" only a minority of a minority of people even had the modems to host or call a BBS. Just by the nature of the home computer market those people would be more technically adept than the average person.
My mom, a non-technical user, can join a Facebook group very easily. She's not going to seek out let alone join some forum even if it's dedicated to the same subject as the Facebook group. She's also not going to run her own forum to talk about some topic where she can simply and easily start a Facebook group.
People seem to forget that in the "good old days" of the early web it was mostly the technically adept building and browsing sites. In terms of conversations had or bytes transferred the vast majority was on closed platforms like AOL and CompuServe. Even in the "BBS days" (the latter era) Prodigy, CompuServe, and AOL were far more popular than BBSes. Even with a BBS being "free" online services had a national reach and just far more resources available. Unless you had a big multi-line BBS in your area dialing into a board could be a crap shoot.
The olden days were not necessarily better than today despite nostalgia and fetishization. Some stuff today is not better than things in the olden days. I'm not saying I like or support Facebook or Twitter or that centralization is unalloyed good. Centralization doesn't just happen in a vacuum and for no reason. Usability is very important as well.
Your assumption of current social media is the only way for mass to run content is not necessarily true. One alternative I would imagine is hosting companies to develop one-click solutions for grandmas using cellphone to create their own websites / blog under their own domains. They can choose whether to allow strangers post comments on their websites.
And internet should be back to BBS days where public content are moderated by volunteers in each community separately?
Now I think section 230 is not a necessity for an open internet. It just changed how internet works by making centralized business easier.