Now in your scenario we can replace x with "help from an organization"
You would think that the groups would look like this
ax, ax, ax, ax, ax, ax, ax
And b, b, b, b, b, b, b
But that isn't quite true. Firstly not all members can receive x based on their membership with group A. Each one of those programs required a secondary characteristic. And the benefit isn't actually help from the program but what that help adds such as qualifying for a loan or advertising. We realize that these programs are put in place to help offset a balance that is already in play. So prior to the help from the program the data set may have appeared as:
ax, a, a, a, a, a, a
And
bx, b, b, bx, bx, bx, b
With the program the results are
ax, ax, a, a, a, ax, a
And
bx, bx, b, b, bx, bx, b
So was the "help from a program really a privilege? It may have been. But when we look at the results we see that Group A is and group B do not have the same level of disparity. And as far as obtaining x, B is not less likely to obtain x than A.
Consequently, i am not sure we can say that the programs you have described are privileging group A.
Like all statistics how you divide the groups and which sets you analyze will allow you to make different statements of truth regarding those sets. But these are not statements of truth about the individuals within those sets. You made the statement that all groups have moments or positions where there group membership brings an advantage or disadvantage. I agree with that sentiment. But that doesn't mean that I think all groups have roughly equal advantages and disadvantages. I think some groups are more advantaged in areas that correlate to our cultural views of success. I think that some groups have more advantages in areas that correlate to our cultural views of justice. I think that some groups have more advantages in areas that correlate to our social views of freedom.
I think that we should do our best to address racial disparity. While racial disparity in itself is not a clear indication of racial privilege it is a clear clue that something is off. Rarely are issues ever so one dimensional that race is the only factor. But, i think people often use this to try to suggest race is not a factor.
On a side note:
"Race" and "racism" carry a lot of baggage as terms go. Ultimately race is a somewhat arbitrary term that doesn't really lend itself to good categorization. However, race and concepts of race are so ingrained in our culture, because of our history, that people cannot help but perceive it, rightly or wrongly. Thos perception of race impacts individuals in our society. Listening to how individuals have been impacted by perceptions of race is helpful in understanding each other, it is helpful in making hypotheses about areas for amelioration. Statistics are another tool to help us toward this same goal. Privilege at its core is just a term that was developed to progress our conversation and actions of amelioration. Unfortunately, i think the term has backfired. Maybe that is because people struggle to have honest conversations about race?