I'm not sure I agree with that framing of the relationship between Section 230 and free speech.
For reference, here's the law:
> (1) No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider.
> (2) No provider or user of an interactive computer service shall be held liable on account of— (A)any action voluntarily taken in good faith to restrict access to or availability of material that the provider or user considers to be obscene, lewd, lascivious, filthy, excessively violent, harassing, or otherwise objectionable, whether or not such material is constitutionally protected; or (B) any action taken to enable or make available to information content providers or others the technical means to restrict access to material described in paragraph (1).
Section 230 was written to solve a very specific problem: Prodigy tried to moderate content on their site, and when someone posted libelous content and they didn't remove it, Prodigy was held legally responsible. CompuServe did not moderate content, and when someone posted libelous content, CompuServe was not held legally responsible. There was a perception that this was a counterintuitive result, and so Section 230 patched over it.
This has nothing to do with the ideological content of the communications. The messages in both cases were already unlawful because they were libelous - the question is whether CompuServe and Prodigy bore any liability (i.e., any obligation to not republish it), or just the end user.
Also, as written, Section 230 does not create an obligation to do anything. You don't have to moderate obscene, lewd, etc. content. You can choose not to moderate anything. The law simply says, 1, you the website operator aren't responsible for what people post, and 2, you don't gain any additional liability if you choose to moderate these things. It doesn't create any liability for not moderating them. The perception (which seems to have been empirically correct) is that Prodigy's approach would be more popular in the market than CompuServe's, and so the law should not create a legal incentive to act like CompuServe. The new law simply removed that incentive; it did not create a legal incentive to act like Prodigy.
The results of the two cases are only counterintuitive if you believe it is good for society for service providers to proactively moderate speech that is already illegal and err on the side of over-moderating. I don't think that belief is easy to reconcile with a strong pro-free-speech view - you're trusting a platform to be making decisions that would otherwise be made by courts, and you don't have nearly the representation/recourse/etc. you do with the legal system, if they decide to moderate you.
In particular, adding an obligation to protect free speech means that providers can only moderate content if they're confident it would result in legal liability. If they're not sure (suppose that, to pick a recent example, someone says that J. K. Rowling "cannot be trusted around children" - is this libelous, or a constitutionally-protected opinion?), they should err on the side of not moderating. But that matches the status quo ante Section 230. If you think that forums should err on the side of under-moderating, then it was perfectly fine to be in the legal situation where Prodigy's approach was riskier than CompuServe's.
Note also that neither of these scenarios does anything to discourage people from running forums where they tightly control what is said (ideologically or otherwise). If I want to host a personal blog with only my own posts, I can do that today, I could do that before Section 230, and I can do that essentially regardless of anyone's proposals (because I have a First Amendment right to say what I want and only what I want). If I want to invite my friends and only my friends to comment, I can do that too. If I want to invite the entire world to comment and I screen comments before posting, I can do that too (I also have a First Amendment right to free association). I'm still liable for unlawful posts (from libel to copyright infringement to whatever else), but if I'm willing to tightly moderate content, that's okay.
Another pro-free-speech opinion here, by the way, is that the real problem is with libel laws, and neither CompuServe nor Prodigy should have been held liable because the speech shouldn't have been illegal in the first place. This is entirely orthogonal to the "free speech" concern of perceived ideological bias.
It's only in the weird intersection of all of these things that the framing of Section 230 and ideological bias seems to make sense - you'd have to take the anti-free-speech view that ruinous penalties for libel are good, and then carve out an anti-free-speech exception that says that if you choose not to exercise your right to say what you want or associate with who you want, libel laws don't apply to you. And then, somehow, the two anti-free-speech approaches cancel out and turn into a free speech view - platforms are obligated to be non-ideologically-biased (in a sense defined by the government) for fear of arbitrary civil penalties.
(By the way, any free-speech reform to Section 230 really should start with repealing 230(e)(5), where FOSTA/SESTA partially removed Section 230's protections so that platforms became responsible for messages posted by users about "the promotion or facilitation of prostitution.")