Your phone isn't secretly listening to you, but the truth is more disturbing

> There is no easy way to close this privacy opening

Sure there is.

Hide screenshot taking behind permission and slap down hard apps that refuse to operate without them.

It says "screenshots of themselves". The application is responsible for rendering the screen in the first place so it fundamentally doesn't need a permission.

Now, what could reasonably be a permission is "access the internet", but our overlords don't approve of that thought.

(Contrast this to web pages, which do not render themselves and thus can sensibly be blocked from screenshotting)

Doesnt android already have a "network" permission? On some roms you can enable it/disable it on install of the app even