First, the headline is a "duh" headline. Of course the lure of robots is the ability to do without humans. That's the whole point -- the very defining characteristic -- of a robot. To automate complicated work.
Second, you assume an all-or-nothing future. That will not be the case. If you have a few more robots and a few more workers, you can drive unit prices down and pick up a few more customers, even without the recently dismissed human workers that can no longer afford to be those customers. That's true at every tier of production, from commodities to luxury goods. This drops the price of labor, and concentrates wages more in the (relatively) few jobs that can't yet be automated. What you see is further stratification of the economy, not collapse.
In your scenario, we're all too poor to be the economy going, and so the economy will never let us get to that point. In a more realistic scenario, a very large portion of the people are out of work, many more have dropped from middle- to lower-class, the wealthiest experience no change -- all while robot labor allows, say, 2/3 of the previous customer base to support the same level of profit for a given product.
10% unemployment is such an absurdly high number that it produces a lot of civil unrest when it actually happens. 30% unemployment would mean a complete re-structuring of society as we know it. The invisible hand proposes the stability of the market, not the stability of our place or our civilization's place in it.