To the extent you are doing something to counteract the negative effects, the system is not capitalism.
This scenario has been covered by quite a few "post-work" authors, who look at humans returning to a hunter-gather society. See Hans Moravec in _Mind Children_ for example.
We used to walk around and these marvelous automated machines produced everything for us: nuts and fruits from trees, free for the picking. In the future, we'll walk around, and robots will be churning out the equivalent of fruits and nuts from factories.
The modern concern of labor arises from the industrial revolution.
What would make it growthist, is if the welfare is paid for by borrowing, which would require compounding interest to pay it off. That the payback requires interest (which compounds, i.e. grows) implies growth in the overall productivity of the system to continue functioning.
Moreover, if you consider these post-work humans to be "poor" something strange is going on in your head. It's easy to conflate "unemployed" with "poor" because that's generally the case in our society. But I wouldn't consider the humans in your "limit case" (with my growthist modification) to be the poor people being screwed over by the growthism. It's the robots.
Strictly speaking, it requires growth in the overall rate of production, which is not the same as growth in productivity (which is production per unit of some input consumed, usually labor-hours.) If, for instance, the rate of population growth were greater than the interest rate on the debt, you could have constant productivity with employment levels without a problem.
Fair point. I'm writing too quickly for my own good, it seems.
And in fact, if you arrange things right, the robots who have the product that is no longer in demand can lend their unused capacity to robots making a product that is in higher demand.
My point is, borrowing from the future isn't always wrong, it depends on the context. If Western civilization had never invented banking or finance, we would have been far worse off.
As robots and computers take over more of what humans once did, we'll have to deal with this problem (see various articles by Kevin Drum). Unless you're fine with a "Mad Max" scenario, some kind of welfare state or minimum guaranteed income is probably the least-worst option.