Both the right to privacy and the right to protection of personal data appear to have pretty big exemptions for government.

The right to private communications was modified by the ECHR to give an exemption for prevention of crime/protection of morals/etc.[1] and the right to protection of personal data exempts any legitimate basis laid down by law[2].

I imagine they'd be able to figure out some form of Chat Control that passed legal muster. Perhaps a reduced version of Chat Control, say, demanding secret key escrow, but only demanding data access/scans of those suspected of a crime rather than everyone.

Legal rulings also seem to indicate that general scanning could be permitted if there was a serious threat to national security, so once a system to allow breaking encryption and scanning is in place, then it could be extended to what they want with the right excuse.

[1] https://fra.europa.eu/en/eu-charter/article/7-respect-privat...

[2] https://fra.europa.eu/en/eu-charter/article/8-protection-per...