Having cloned its core millions of times over, the AI is now a civilization unto itself.

Precisely. It is then a civilization, not some single monolithic entity. The consumer PCs have a lot if internal computing power and comparatively very low inter-node bandwidth and huge inter-node lag, entirely breaking any relation to the 'orthogonality thesis', up to the point that the p2p intelligence protocols may more plausibly have to forbid destruction or manipulation (via second guessing which is a waste of computing power) of intelligent entities. Keep in mind that human morality is, too, a p2p intelligence protocol allowing us to cooperate. Keep in mind also that humans are computing resources you can ask to solve problems for you (all you need is to implement interface), while Jupiter clearly isn't.

The nuclear war is very strongly against interests of the intelligence that sits on home computers, obviously.

(I'm assuming for sake of argument that intelligence actually had the will to do the conquering of the internet rather than being just as content with not actually running for real)