This doesn't explain how they'd carry out such a plan with AI specifically, or why they'd be motivated to do so in a way that is different from past motives surrounding mass murder.

Genocide is already a thing long before AI. The victims of colonialism have been dealing with existential threat for hundreds of years. Some peoples in the world are currently under existential threat right now, not because of AI, but because of already-existing murderous imperial policy.

Like nothing personal, but your narrative is fudging a hell of a lot. For example, what does "excess population" even mean? Who defines it and for what reasons? Billionaires suck the same as other exploiting classes before them, but one thing that they are not, is illiterate about power. When a colonial king mutilates the people in the Congo, it's not just cause he's feeling murderous, but for material gain, control over resources and labor there.

Or another example, the empire has been mass murdering those who refuse to submit to imperialism for a long time now (without AI). AI is not going to magic away the empire's attributes of decline, like outsourcing industry, financialization of the economy, grifting to feed on its own imperial machine. These are things that have been in motion for decades with many moving parts involved. They cannot be reversed overnight. MAD nuclear doctrine is the closest thing to what you describe and that has been in place for decades, long before AI. And so far it hasn't happened. It still could, but much of the world population being dead and its environments being uninhabitable wouldn't be very good for business, so even the more vicious capitalists are unlikely to be thrilled about triggering such an event.

I think they meant "how" as in "how is it possible for a bunch of computers which have no actual intelligence and which are completely dependent on humans kill all humans?" and not who would make that happen or why. Like, the actual mechanism or chain of events that leads from computers to extermination.

But they can already just tell a buman to do that

Yep, once the billionaires can use AI to solve the worker-capitalist conflict by replacing all workers, they will make the former workers go the way of work horses after the automobile. Much easier to kill us all with Terminators than to continuously pay us a UBI.

Either way, we will get to some form of communism. It will either be communism for everyone, or communism for just the billionaires with everyone else dead.