The calculus is simple:

- People who go into government want power over others, and they'd therefore prefer more power than less at every junction

- The population put up very little resistance in general - e.g. as we saw with COVID, they'll freely let the government put them under house arrest and destroy their businesses, to protect them against a bug only slightly deadlier than the flu

- However they do need some pretext - with none at all, impinging on people's civil liberties would cause confusion, and the people might even ponder grumbling about it, which would be unnecessarily messy for everyone involved

- Hence they simply need to lay out some half-assed pretext, just to make it easy for people to internally justify their further loss of rights, so that they don't have to have any uncomfortable thoughts like "maybe I should disagree with this?"

- "Think of the children" is a nice easy one. It's basically a farcical meme at this point, but that doesn't affect its effectiveness, for the reasons described. You could use this for literally anything, no matter how extreme (I'm sure for instance the Nazis used it as one reason for exterminating Jews - to protect "Aryan" children & their future etc)