I respectfully disagree. I have known, in a variety of settings, a wide range of white women. Poor ones are more overtly racist, as a rule. Poor and middle class white women do not strike me as any less racist than wealthy ones, though they are often more straightforward.
Whiteness is an advantage, no matter who has it. Cooper's behavior is an example of white privilege.
White women are the only women on planet Earth who have white privilege. When they do not acknowledge it, regardless of economic status, they nearly guarantee it will be replicated. ALL OTHER WOMEN LACK WHITE PRIVILEGE. This makes white women unique in that they have an advantage the rest of us don't have.
Yes, wealthy people have more economic privilege than poor ones. This may be temporary, as it is possible to move between economic levels.
White people, regardless of wealth, have racial privilege that POC"s don't have. It is permanent. The corollary is that non-white people will NEVER move between white and non-white, no matter what. White people are not just privileged, they are permanently privileged.
White women have racial privilege at every socioeconomic level. If they continue to turf this, they will have little leg to stand on when it comes to criticizing patriarchy.