Their benefits come in the form of “Jobseeker’s Allowance” (unemployment benefit), income support and “incapacity benefit”, as well as a disability allowance for Mrs Chawner’s asthma and epilepsy, a result of her weight. Additional money for conditions such as heart disease and diabetes top off the largesse.
So, although we're calling it "the dole", they're actually getting separate checks monthly from at least 5 sources. Their weight and shape(s) aside, I'd say that's administrative talent. They could be sitting behind desks processing and coordinating disability, unemployment, and income support bennies for the entire population, not just themselves. "Community Benefits Coordinator". You'd think that such work would bring home more than they're taking in now.

Although I grant the point: what's their incentive?

IMO: gov't should insist, since the family (and thousands like them) are de-facto gov't employees on retainer, that they perform some public service.