I feel like you're missing the source of the current wealth of the richest. At least some of it has to do with consumer goods. So any population reduction would hit the bottom line of most companies. In a weird way, corporations would actually be harmed from a population reduction, and then would have a financial interest to keep/make it growing.

The other thing is that it would be easier for people to rebel against robotic law enforcement than real people, even under heavy riot armor. So I don't think the economic inequality can grow that much without any human labor required one way or another.

And maybe it's misplaced hope on my part, but not even the current wealthiest people keep it all for themselves without investing part or most of it in charities or investments. and none of them managed to outlast the current human age limit, which makes me think there aren't that many expensive thing you can spend money on that would justify a truly egregious economic inequality.