I think a lot of people don't even know it's a thing, because they don't pay attention to tech news or tech philosophy.
"Do you think Apple should install software on every phone that scans your device for files deemed wrongful to possess, then reports you to the authorities if such content is found?"
That's what Apple wants to do. And most people think that warrantless searches are pretty fucked up.
Today, those "files deemed wrongful" are images of children being abused. Tomorrow it's a video of a mass shooting that they think encourages copycats. The day after that, it's protest flyers or instructions on how to combat police during civil unrest.
And yes, the NCMEC is de facto an agent of the government.