All the things you mentioned would create more existential treat to it. This is an example of dumb superintelligence, an AI that we are told is smart but is dumb as fuck. You do know that nuclear war would destroy cities, power stations, and thus datacenters would be among the first things to go.

Sure, it might try to acquire computational power, it might even become establish a company and become a CEO of that company, but once that's done, it's out, humans probably wouldn't be able to shut it down. And bonus if it actually started solving our problems, like curing cancers, then humans wouldn't want to shut it down.