I haven’t read the whole Barnardos piece (and I have no idea about “woke”) but I do see why you would be offended. They write
For the one in five Barnardo’s service users who are Black, Asian or minority ethnic, the colour of their skin is an additional factor that negatively affects them and their families in a multitude of well documented ways.
I think that’s incontrovertible but to then say that the 80% is benefitting from “privilege” (white or otherwise) is a complete misuse of English and certainly offensive!
Among the 80% there will no doubt be some/many who have additional disadvantages (disabilities for example) that would make them less “privileged” in Barbados terms, and ditto among the 20% BAME
. Just not helpful language.