I don't necessarily agree. That could be an outcome, but it would require policies or voluntary restraint to get there.

At least in my country (Uruguay), the lower earning 8% have 30% of the children, placing a huge strain on society and services (I don't see robots replacing medical and social services anytime soon).

So, to enable a sustainable society like you propose, something like a two children per couple rule should be enacted (hopefully nothing so harsh as the Chinese version).

If these benefits are demanded, either by government action or organized labor. If it's up to those who control corporations, they have no reason to cut hours and increase benefits out of the goodness of their own hearts; they'll just fire people.

It would be great if that were the result. But it remains to be seen. The requirement is that humans decide to change their economic system.

If we don't do that (and human society is really good at staying on autopilot until problems have gotten truly severe), we'll just end up with a mostly-automated society where the robot owners have everything and everyone else lives in a ghetto.