I worked with welfare recipients in the 80’s and 90’s at several state unemployment centers. Here’s a typical arrangement, from what I saw. Mom gets TANF (used to be AFDC), food stamps, WIC, possibly Section 8 and government cheese and she’ll work an off-the-books gig of some kind. Dad, if there is one, is on unemployment, workman’s compensation or some kind of disability. He also has an off-the-books job or side business (legal or illegal) and also often works for goods or barters things, etc. The kids all get free school lunches and are often on disability. The family goes religiously to several food banks and receive thanksgiving, Christmas and seasonal clothing from various charities.
My experience as well and you perfectly described the situation. You must have been working with a better selection than I as the primary source of off the books income for the people I was working with was drug dealing, prostitution and robbery.