To qualify for safe harbor (as a service provider) under the DMCA, you must register a designated agent at the US Copyright office: https://dmca.copyright.gov/osp/publish/history.html?search=noauthority&id=06a0e40b2f20b4ccdcc5e18ba7e4af1d
This is unrelated to Section 230.
If Moon v Greer goes to the supreme court (unlikely), I suspect Moon will lose not only because of his actions (and inaction) in that particular incident, but because he failed to register his website with the US Copyright office and pay them the $6 extortion for safe harbor.