Yes, but it’s harder to game. Income is vastly more fluid than racial identity, especially since in this situation it’s more advantageous for your income to go down.
Income and wealth are better predictors of “privilege” than race though.
So one is using a worse indicator.
Using a mix of income and wealth over several years to me is an appropriate indicator and cannot be easily gamed.
The fact that Will Smiths kids get a better chance to get into college than a similar applicant from a first generation immigrant family from Vietnam shows how broken race as an indicator can be.
How is income vastly more fluid than a “racial identity” checkbox on a form? At least it’s sort of possible to verify.
It’s easy to lie and claim to be black, nobody will perform tests on you to verify this. You can be the whitest person in the world and get away with this (so many do).
Just because it can be gamed doesn’t mean it’s not a good metric