They pitch the above mentioned privacy assuming you abide by their ToS and the laws of their land.
When the authorities issue a notice to take down illegal data, the cloud providers have to comply - track down and remove the data from their servers. The recent CLOUD Act further clarifies situations where the US government can issue takedown notices on US cloud providers even if they host content overseas. (Of course, there are provisions in place for cloud providers and citizens to challenge these. Note, CLOUD Act principally focuses on surveillance, i.e. FBI requesting data about an individual using the services, but it extends to removal of illegal data as well.
Well, what you are stating is the capability of LE to know that illegal data is in the cloud means that the clouds aren't secure at all, and that the providers are completely aware of all the data in the cloud, and so is LE.
Do I understand correctly?