I'm not American, so same situation, but welfare as a cost to government I am pretty sure includes all spending, not just that which goes to the final recipient. https://www.budget.senate.gov/imo/media/doc/CRS%20Report%20-... is the best I could find.

I'm also pretty sure that is the case in almost all countries, where welfare is all money spent on, well, looking after people in some way?