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.
Section 512(c)(2):
(2) Designated agent. – The limitations on liability established in this subsection apply to a service provider only if the service provider has designated an agent to receive notifications of claimed infringement described in paragraph (3), by making available through its service, including on its website in a location accessible to the public, AND BY PROVIDING TO THE Copyright Office, substantially the following information:
(emphasis mine)
...