the 1935 social security act excluded industries which relied heavily on Black labor. 65% of the Black workforce was excluded from the initial Social Security program (and only 27% of white workers)
during the Great Migration, Black people began moving north and into urban centers, and started accessing welfare in higher numbers, which results in increasing hostility towards welfare programs
in the 60s, the welfare movement also begins to become associated with civil rights, with a Black, urban face of poverty, rather than the white, Appalachian face of poverty that was more common previously. This further adds to the growing backlash against welfare programs.

https://www.newamerica.org/weekly/rise-and-reign-welfare-queen/

Most Americans see Black people as the face of the welfare system. Black people and undocumented immigrants are generally considered to be the largest welfare recipients. This is not true.
white people are less likely to support welfare when told that Black people might benefit from them

https://www.vox.com/policy-and-politics/2018/6/13/17460362/race-food-stamps-snap-farm-bill-2018-republicans-welfare

https://www.vox.com/2018/6/7/17426968/white-racism-welfare-cuts-snap-food-stamps

Clinton's 1996 welfare reform gives states greater latitude in designing their own welfare systems. As you might imagine, this results in southern states with the highest Black populations dismantling their welfare systems and paying out the lowest benefits in the nation.

https://jacobin.com/2016/10/clinton-welfare-reform-prwora-tanf-lillie-harden

https://www.thenation.com/article/archive/why-it-matters-that-hillary-clinton-championed-welfare-reform/

https://www.theroot.com/when-states-run-welfare-black-children-are-the-ones-wh-1790854181

So, opposition to welfare is rooted in a history of racism. Most Americans, when they get upset about welfare recipients, are picturing an urban Black person. And welfare reform has historically been used to take benefits away from Black people. You might not think you're talking about race when you complain about welfare recipients or talk about welfare reform, but you can bet your ass most people in this country are. To complain about welfare recipients and talk about welfare reform (unless you're talking about expanding welfare) is to talk about race, even if you really are so innocent as to believe that you're not. For this reason we are talking about the song having "dogwhistles."