I don't understand why the dishwashers leaving needs to cause societal breakdown, though. After they leave, surely someone thinks "Hm, we have no clean dishes any more and nobody's volunteering, so I guess I'll need to clean some dishes, if only for myself, but then once I'm in dishwashing mode it's efficient to clean more dishes than I personally need, and even better if someone decides to reward me for cleaning their dishes by doing my laundry since they realized someone needs to do that".

I mean, isn't this "the invisible hand of the market" at play, just without the intermediate medium of exchange?

It is true, though, as the end of the article points out, living this way defies pretty much any kind of planning, and that can be scary for people.