White "privilege" does exist, the problem is the language democrats use to describe it. When I was growing up (back in the 80s) what was discussed was minority disadvantage. If you try to eliminate disadvantages, you're bringing people up, if you try to eliminate privilege, you're bringing people down. Also, privilege implies special treatment, meaning the term inherently implies that minorities are being treated fairly, it's just that whites get special perks.
For instance, when someone tried to explain privilege to me, they brought up that I had a two parent household. I thought, "that's not a privilege, that's supposed to be the baseline."