Automation is good. Eventually, it will lead to shorter working day (e.g. 6 or even 4 hours), shorter working week (e.g. 4 days, or even 3.5 days), to longer vacations, to better health care, earlier retirement, and so on.
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 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.