The poor won't die in the apocalypse leaving only the rich behind. The poor will die, and the rich will be faced with the harsh reality that they needed an army of poor working under them to sustain themselves, leading them to all die within the generation.
You still need loaders, drivers, retailers to get anything to the customer. A lot of rich ski and holiday towns can't staff the stores and Cafe's, because the employees can't afford to pay rent in the same towns, so they face a similar issue...
Maybe, but there are still a lot of invisible people involved to get the food all the way to your table. And small suppliers cannot afford to switch their whole operation to robots.
just to nitpick, they said the "poorest", not the poor in general.
The poorest are the most vulnerable, and, I suppose, not absolutely crucial for the ruling class.