My 50 year high school reunion is this summer. You apparently have no idea of the changes that came about during my lifetime and Carver's lifetime. Her brothers are probably my age, or even older, and I remember separate bathrooms and drinking fountains in the south. I remember seeing the protests and marches and firehoses on the nightly news. I was 9 when the Civil Rights Act was signed into law by Johnson. That changed things on paper, but only partly in practice. And I very clearly remember MLK Jr's assassination in 1968.
I see no reason to doubt what Carver said.