Of course they care about it in the sense that they care about maintaining law and order which forms the foundation of their power. Everyone knows the emperor has no clothes when it comes to "think of the children" bills. It's a fig leaf over their power grab. Everyone also knows that this will ultimately be used to crack down on dissidents, criminals, and other undesirables (likely immigrants as well) and that is where you'll find the democratic constituency for this overreach:

Authoritarians generally and right-wing populists in particular are the major proponents of government surveillance and control.