They are conservatives. In Germany they also try every time to enact Mass Data Retention ("for catching Criminals"), then the courts decide it's not compatible with the constitution, and after a few years they try again.

I highly doubt they have given up here too

I see the exact opposite. Let's classify parties:

EPP (Christian-democrats, ie. centrist to somewhat conservative parties): >95% voted against. These mattered more than any others

Patriots for Europe (far-right, even racist): 2/3 voted against.

"Green"/progressive parties: ~90% voted in favour

Socialists from anywhere on the spectrum (from just-barely-left-of-centre to communists): >95% voted in favour

Laissez-faire parties (meaning rightist, but not racist): largely voted in favour.

https://howtheyvote.eu/votes/189574

Leftists and Green parties desperately seem to want ChatControl (not just this vote), against literally everyone else, but especially against centrist parties.