Eg, someone could use the phrase "Won't someone think of the children?" to describe a legitimately bad thing like bank fraud, but the solutions that flow from the problem that "children are in danger" are significantly different from the solutions that flow from "phishing attacks are rampant".

The two issues in this case aren't quite as different as child-endangerment and bank fraud. But if the problem was as the original title describes, the solution is quite different (better sandboxing) than what the actual solution is. Which I don't know, but better sandboxing ain't it.

'ignore the facts! ENEMY!!!' generally doesn't end well for anybody

Like OP, I don't consider behavior confined to the browser to be my computer. "Scans your browser" is both technically correct and less misleading. "Scans your computer" was chosen instead, to get more clicks.