What do you do with the majority of the population that doesn’t take care of the bots? Starve them? Feed them for free? Turn the bots on them?
In a free society, like the early Republic of Rome, suddenly gets flooded with cheap labor (in that case, slaves from Germany) that the free man can’t make a living, you have a situation that will either lead to revolution, or to the State caring for most people in order to keep them happy. In Rome, it led to both. Massive welfare and a revolution from a Republic to an Empire.
We are trending to the same place.
What you are describing is a political problem not an economic one. There can be no inevitable march towards the obsolescence of man unless the government creates the conditions for it.
I can see how that might happen here -- but it is not economic law. Once again, it will be the work of politicians.