You answered your own question when you said:

The AI is going to stupid, but it's going to quickly find out how to turn the world into paperclips. It's not going to be a general intelligence. But it doesn't have to be to cause problems.

Incorrect. It very much DOES have to be a general intelligence, and far from stupid, if it is going to be smart enough to evade the efforts of humanity to squelch it. That really is the whole point behind all of these scenarios. It has to be an existential threat, or it will just be a matter of someone walking up to it and pulling the power cord when it is distracted by a nice juicy batch of paper-clip steel that someone tempts it with.

Or, as Rick Deckard might have said:

"If it's an idiot, it's not my problem"