I think the challenge for society here is not to simply reject attempts like this, but how to prevent them from being pushed over and over until a specific context allows it to be approved.
I'm convinced the people suggesting this type of thing are influenced or even compromised by their constituent's enemies and NOT the result of poor education on the topic. This policy for example would be most helpful to enemies to the EU. It would lower the cost of acquiring the data for China and Russia as it allows them to mass acquire data in transmission without incurring the cost of local operations. The easies…
Yes, it would lower such barriers for countries that are commonly seen today as Europe's adversaries. But in this case, the U.S. (or rather, U.S. organisations and corporations) might be the primary bad actor pushing for ChatControl. See e.g.:
Thorn (organization) - https://en.wikipedia.org/wiki/Thorn_(organization)
"Thorn works with a group of technology partners who serve the organization as members of the Technology Task Force. The goal of the program includes developing technological barriers and initiatives to ensure the safety of children online and deter sexual predators on the Internet. Various corporate members of the task force include Facebook, Google, Irdeto, Microsoft, Mozilla, Palantir, Salesforce Foundation, Symantec, and Twitter.[7] ... Netzpolitik.org and the investigative platform Follow the Money criticize that "Thorn has blurred the line between advocacy for children’s rights and its own interest as a vendor of scanning software."[11][12] The possible conflict of interest has also been picked up by Balkan Insight,[13] Le Monde,[14] and El Diario.[15] A documentary by the German public-service television broadcaster ZDF criticizes Thorn’s influence on the legislative process of the European Union for a law from which Thorn would profit financially.[16][17] A move of a former member of Europol to Thorn has been found to be maladministration by the European Ombudsman Emily O'Reilly.[18][19]"
Additionally, it would not surprise me at all if Palantir is lobbying for this either. Many EU countries, like Germany and Denmark, have already integrated Palantir's software into the intelligence, defence, and policing arms of their governments.
But at the end of the day, while it is convenient to blame external actors like U.S. corporations, ultimately the blame lies solely on the shoulders of European politicians. People in positions of power will tend to seek more, and I'm sure European politicians are more than happy to wield these tools for their own gain regardless of whether Palantir or Thorn is lobbying them.