The historical purpose of safe harbor was to allow AOL to have a walled garden without become responsible for all the illegal content that snuck past.
IANAL, but DMCA safe harbor doesn’t really apply to a companies like CloudFlare, nor is it needed, because they are not moderating UGC on a publishing platform, they are a routing optimization utility.
Content moderation is covered by the CDA section 230; completely separate from the DMCA
And it 100% should apply to Cloudflare as well; if they're caching copyrighted material they need to respond to takedown requests or lose that safe harbor protection. Just as it applies to Google for indexing alleged copyright material; if you've ever seen that "links removed due to DMCA takedown notices" in search results, even when they are not hosting said content.
IANAL also, and I don't know how much DMCA stuff is handled cross borders (I do believe some DMCA provisions are applicable thru trade treaties and the like, but not sure); and I do think DMCA takedowns are rife with abuse; but content moderation really has nothing to do with it..