It's an interesting approach ... but i see a few issues. To make it work ... specifically this part "undertaking work of public benefit at the minimum wage."
You need a minimum wage that a single person can live on at 35 hrs per week...
And by "live on", I mean eat healthy, have a warm safe place to sleep in a proper bed, have sufficient climate appropriate clothing, provide the incidentals of life(toiletries, laundry detergent etc), have enough to pay for communication and transportation costs sufficient to be employable and pay for healthcare.
Even using very conservative numbers (8k housing, 3.6k food, 0.5k clothing, .2k incidentals, 1.2k trans & comm, 3.2k Medical = roughly 17K) the federal minimum wage would need to be set at least $2/hr more than it currently is. And that assumes they never take any time off working.
And if they have to work all the time just to get the minimum income they need to survive, how is that significantly different from debt peonage?
So then ... let's raise the minimum wage to $15/hr and set it to automatically adjust with inflation. This increases earning to the point where people on the jobs program can actually live fairly well and save money.....
But how does it work for people incapable of working? Wouldn't you need 2 programs and all the bureaucracy needed to determine which program a given person falls under and all the costly checks to ensure no one is cheating the system?