When I taught students about white privilege, “white privilege only applies in cooperative — and not competitive, situations” — is not what I taught them. Many privileges relate to finite resources. You have only so many spots at the best law school. You have only so many jobs available at Google. There are only so many Senate seats. These resources have absolutely been coveted by race privilege.
Giving up white (or gender, etc.) privilege necessarily means you take on the same risk as every other person without the benefit of an advantage. Now your odds of accessing finite resources go down. We can’t tell people that equalizing society and taking away automatic advantages from a group won’t impact that group. It’s going to. The question becomes whether that group has a sense of fairness greater than group differences or whether they can come to see the group as part of their larger group. The second question of self-interest to the majority group member is what do I gain by not being assured I will be the default culture? I thought that taco truck on every corner idea sounded fantastic (I’m presuming you heard Trump’s Latinos for Trump President).
Right now, white people can’t shed their privilege. The best example is gentrification. A lot of the well-meaning white people moved to inner city communities as part of an effort to integrate — only to make those neighborhoods unaffordable to their black neighbors in the process. They can’t build more homes. The space is filled, so the property values rise.