Thought experiment. Two crazy looking homeless people approach you yelling. They are dirty in ripped clothes and look to be high on strong drugs. One is black, one is white. Do you feel better about, or differentiate the white person because he is "your people"?
Homelessness is a problem that must be solved at the highest possible level of government. Since the 'problem' is able to move around the entire country that is the only way to fairly have the collective whole (society) integrate and restore the individual in need to either being a productive member of society, or at least a dignified living solution for when medicine or some other aspect has failed.
Seen in an even wider view, much of illegal immigration is quite likely driven by the homeless or persecuted ignoring the borders of nations that have failed them, as they seek the natural birth rights of anyone on their planet.
Unconscious biases are likely to have much more effect in ambiguous situations where it's harder to try to understand someone's motivations or how to react, than in a situation where those things are clear. Their absence in the latter does not imply their absence in the former.
To continue, briefly. The difference between the various other types of benefits vs "whiteness" is twofold but interconnected. It is obvious who is and is not in the group and the group has institutional power. Most other sets of parameters do not work together in this way.
Again this isn't my position so if it is yours and I've misrepresented it, chime in please.
Regarding the point itself, rich people or higher classes people as a group also have "institutional power", much more than white people as a group. It is pretty easy to differentiate people background after some sentences as the research shows. So the only difference between those two disadvantages is exchanging some words. Even if it is a significant difference, it still doesn't change or cancel other circumstances in a person life and doesn't make this difference overruling any other difference.
The idea goes something like this. You have white privilege because you are visibly white. By being visibly white you gain an automatic differentiation from those who are non-white. The belief is that the system, being largely run by white people will extend courtesy and nicer treatment to those who look like it whilst those who differ in appearance must "earn" the position you are given by the elements of your birth. It's a bit more complicated than that but that's my understanding of the core of it.
I am not trying to strawman the argument and if I have it wrong, please explain where I wandered off course.