I agree with you about the effect of welfare but your example of the military supports my take.
Black men joining the military were very limited compared to whites. They were segregated, they couldn’t be officers, in the navy they were cooks and stewards not machinists or gunners. During WWII the military opened spaces for them because the manpower was needed, after, they were integrated. That’s two generations into the 20th century, two generations delay.
So no.
It does not "support your take" it undermines it and then blows it up.
Let me tell you a bit about my granddad's "white privilege" he was born to dirt farmers in Appalachia the oldest of eleven children that lived. He was just a hair to young for the draft for the Great War. His dream was to go to the "big city" Dayton and get a job as a mechanic. When his mom and dad both died in the Flu Pandemic he stayed because someone had to look after the young ones.
A few years later he married a young lady and they settled down on the farm. Which was lost in the Great Depression when the bank failed. They owed a whole $300 on the place.
They moved to Athens, a collage town, not so he could go to collage, but so he could work towing and fixing cars. My dad never had a second pair of pants until he was in his teens. He suffered to this day from rickets that he had as a child from malnutrition.
Come WWII his oldest brother was conscripted. He was not a officer, or even a cook, he was a diesel mechanic and spent his days covered in grease and grim in the bowels of a ship.
That covers those two generations of delay those poor people of slightly darker hew had to suffer in the 20th century.
Were there poor and under privileged? Yes, of every shade.
My mom's family actually had a worse time then that.
My point is that while you looking back seem to think that if you were lighter then a paper bag you were on easy street and if you were darker all doors were closed you do not know actual history.
Angela Davis' father, Frank, owned a service station, while her mother, Sallye, taught elementary school and was an active member of the NAACP. This is middle class during a time when, according to you, those doors were closed. No, they were not.
They had major privilege compared to my family. She went to collage in Germany.
My family only went to Germany to either shoot Germans or protect them from the Russians.