I wonder if even a term like "white advantage" might be better.
I suspect part of the problem with "white privilege" is that many people interpret the word "privileged" as meaning "wealthy and in a high social class".
Of course that's not what's MEANT in this context, but it's not surprising that people misinterpret it and come back with stories about their dirt-poor Irish ancestors.
(In the same way, "white supremacism" is a really unhelpful phrase, because it conjures up images of the KKK and so on. Again, very likely it's not MEANT to, but that's the way it's widely taken.
There, something like "white dominance" might be better.
But it's probably too late to change the language of the discussion...)