I don't think this is true at all. Twitter employs (employed? https://www.pbs.org/newshour/economy/outsourced-content-mode... ) paid moderators. Facebook does ( https://www.theverge.com/2019/2/25/18229714/cognizant-facebo... ). Plenty of other forum sites have owner-operator mods.
"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." (47 U.S.C. § 230(c)(1)).
That explicitly applies to providers and doesn't say they aren't allowed to moderate at all.