On receiving a DMCA takedown notice they have to remove access to the content. Them's the rules of the DMCA. You can then file a counter claim if the notice is mistaken or fraudulent or you have the required permissions. The next step for the claimant is then to either remove their claim and allow your content to be restored or sue you. The DMCA also requires repeat infringers be banned from the service. In this case I imagine that means your account, not singular website, gets banned.

If they don't do that then Cloudflare loses their safe harbor status and would be liable for the content you upload.

While possibly controversial to say, the way YouTube handles copyright claims is actually far easier and much safer (legally) on the end user.

You should always seriously consider the repercussions if you're going to have a site that handles user-uploaded content. Primarily make it as easy as possible for your service to receive the initial DMCA takedown notice rather than force the claimant to go to Cloudflare directly.

For a similar process check out how GitHub handles DMCA claims: https://help.github.com/articles/dmca-takedown-policy/