I don't think they should be able to claim "safe harbor" protection (or am I getting that confused with common carrier?) and then get to use censorship however they like. The spirit of safe harbor is that "we're not responsible for what random people post on here". It doesn't seem logically consistent to claim this legal protection but then curtail the content to anyone's personal sensitivities.
Section 230(c)(2)(a) is pretty explicit in allowing moderation.
Just because they are not legally liable for what users post, Congress did not want to prevent platforms from going beyond their minimal legal requirements if they so chose.
The alternative us a law that says "It is unreasonable for us to expect you to perfectly police your users; but we will hold you liable for imperfectly doing so".
Just curious: do you believe in net neutrality? (I think that's the cause of my conflation with "common carrier"). But if censorship is a valid reason to deny service, why can't charter or AT&T say "we dont like what you're sending over our wires, so we're blocking or throttling you" ?
Is it merely the fact that encryption blocks the carrier from knowing what the line is used for? Or said another way, should AT&T have the right to terminate a user's service if they were certain a user was posting, say, white supremacy?