Earlier quoted context omitted.
Racial discrimination in the US significant enough that, for example, merely having a "Black" name leads to worse results from job applications, regardless of other factors [1]. Merely focusing on wealth levels won't counteract that kind of effect. [1]: https://www.businessinsider.in/policy/economy/news/distincti...
They never run these experiments right and this one looks to be no different. What they choose as "Black" names are those associated with lower class blacks but they do not do the same with "White" names. In this case, they use names like Latonya for blacks and Heather for whites. You never see them use names like Cletus or Billy Bob or other names associated with lower class whites.
The names in the NBER study were drawn from actual birth certificate data correlated with race. It was not a subjective choice from researchers. Latonya was used because Latonya was the first name given to 4.7% of the black female babies born during the study period. Just like Allison was 4.7% of white female babies in the same data.
If you perceive common African American names as lower class than equally common white names, I believe that's part of what the study is trying to demonstrate and not a flaw of its design.