As I understand it, chat platforms provider will not be held in violation of the data privacy laws if they add automatic detection and reporting of unlawful content to their platforms. E.G. a CSAM detector in a client app for an end-to-end-encrypted messaging service would be lawful.

You're looking for an answer that doesn't exist. The term "Chat Control" was coined by the opponents of these proposals to express their worst case assumptions; they reject the idea that the specifics matter, because they fear that any kind of chat scanning can be abused in basically the same ways. Supporters of chat scanning proposals don't call them "Chat Control" or view all such proposals as part of a unified whole.

The answer is already specific, but not complete.