I read and re-read both the DMCA 17 USC § 512 and Section 230 of the CDA, and as far as I can tell, the DMCA only requires responding "expeditiously" to DMCA takedown notices and court orders, and the CDA has no conditions at all but doesn't protect from copyright liability in the first place.
In fact, the CDA explicitly states that its liability protection DOESN'T require neutrality, and extends to “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”. See https://www.eff.org/issues/bloggers/legal/liability/230