The obligatory IANAL.
But common sense tells me that every case has to be judged on its own merit, and just one particular service cannot “infect” every other service and re-classify the entire Cloudflare company even if Cloudflare does not host and has no control over the service in question.
For R2, the individual files are permanently stored on CF’s servers so, yes, I want to believe a valid legal request can be made for the removal of offending content. For Pages, CF is the host of the entire site… there’s no 3rd-party host involved so the “we’re a network provider” argument will not fly.
Each case is different, that’s why there are different terms of service for different services. Please see: Service-Specific Terms - Cloudflare
In particular, see this from the Developer Platform ToS (covering Cloudflare Workers; Cloudflare Pages; Workers KV; Durable Objects; Cloudflare Queues; and R2):
- Unlike most Cloudflare products, the Developer Platform can be used to host content. Content stored on the Developer Platform (whether in conjunction with a Cloudflare storage offering or not) that we determine in our sole judgment to be illegal, harmful, or in violation of Section 5 of the Cloudflare Developer Platforms Service-Specific Terms may be blocked or removed, and use of the Developer Platform for storage of such illegal or harmful content may result in suspension or termination of Cloudflare Services. While we generally try to provide notice of such action, we reserve the right to take action without notice as appropriate. For these purposes, illegal or harmful content includes but is not limited to: (a) content containing, promoting, or facilitating child sexual abuse material or human trafficking; (b) content that infringes on another person’s intellectual property rights or is otherwise unlawful; (c) content that discloses sensitive personal information, incites or exploits violence, or is intended to defraud the public; and (d) content that seeks to distribute malware, facilitate phishing, or otherwise constitutes technical abuse.
As to whether CF will honour such requests is a different story.