Chat Control is being pushed by national governments, either directly or through the meeting of their leaders, the Council. EU institutions are the ones continuously keeping it at bay.
Chat control is being pushed by national police forces as well as europol. It's... Lobbying. Basically. The whole story of how it started with ylva johannsson is the result of strong lobbying by Thorn and Ashton Kutcher
Where I live, we have and exercise the right to legislative referendum, which stops such legislation in a very clear and decisive way. If something like this passes in the EU, we have no way to fight it (international treaties are not subjects to referendum). The influence in EU parliament is delegated on so many levels that it's impossible to transparently see what your vote influences.