How about we use a different phrase that sounds less like they're supposed to take it personally, then?

You can't escape the connotations of the word "privilege" just because you stick it onto "white" and declare it isn't personal. Not quickly anyway.

For instance, if I say, "Peggy is a PIG!" and by "PIG" I actually mean "pretty interesting girl", it's eminently reasonable for Ms. McIntosh to get really upset when she hears me--and it's my fault for not communicating clearly, not Ms. McIntosh's fault for having acronym fragility.