There's no such thing as common carrier status for CDNs, but, when you're in court trying not to take a site down you'll have to make arguments about why you to

There's no such thing as common carrier status for CDNs, but, when you're in court trying not to take a site down you'll have to make arguments about why you took others down but don't want to take this one down, part of that is how consistently you applied your own standards. "We don't take sites down if the content is not illegal" is a pretty clear consistent position to fall back on. "We don't take sites down if the content is not illegal except that one time the CEO decided we would" is not, so you need a different reason, and they don't have one anymore.