There's no mention of Social Security, Medicare, or the Air Force in the Constitution either. Should we eliminate those?
Welfare comes under the Congressional power to dispose of federal property.
Article IV, Section 3
“The Congress shall have Power to dispose of and make all needful Rules and Regulations respecting the Territory or other Property belonging to the United States”
Instead of given a bundle of money to Binman Bob to throw into the back of a Waste Management vehicle, Congress can given the money to Binman Bob’s sister, who’s a young mother in need of money.