While all safety net programs would benefit from a more localized administration (with guidelines from a national parent program to guard against corruption and austerity mindsets) they should be funded at the federal level. The reason for this is simple. The federal government is not restrained by tax collections as it literally creates any money it needs for the public purpose. It is the monopoly issuer of US dollars while state and local governments are, like us, simply "users" of dollars and dependent upon tax revenue.
With federal funding and local administration the money created would be interjected into local economies countercyclical to local economic activity. As such, it would help shore up those economies as needed instead of becoming a burden on them when they can least afford it. A community undergoing job losses and needing to provide income security for its residents will be much more generous with benefits than one whose residents are already strapped with loss of revenue, as the benefits, including administration costs, would be welcomed to keep vital supply chains intact.