This is some Terminator version of destroying humanity, but a super intelligent AI could be much more insidious, and simple.
It looks a lot more like social engineering.
One easy, obvious example that has also been explored a thousand times: it could convince all nuclear countries that they are under attack by another nuclear country. Everyone nukes each other and Earth enters nuclear winter.
Maybe not every single human dies, but humanity is effectively destroyed, by our own hands!
> Perhaps an agent could take all internet connected services offline
And there wouldn't even be Spotify!
The most likely humanity-destroying outcome is the one we can't imagine, because a super-intelligent AI is smarter than all of us combined.