I think Kad frames the issues pretty well and asks the right questions--or, at least, the questions I'm asking myself.
"But that malleability is problematic for a lot of people, since it erases established markers of identity."
I think another issue, certainly for African Americans and very likely for women (though I'm neither, so I admit I'm making assumptions here) is the notion of taking on the mantle of each group's historical oppression. It's probably dangerous to play the "if ______ were ______, we'd feel differently," but I'm guessing that if a black person were assuming a white identity, we'd be less worked up about it as a country.