* Spoiler Alert *
I think you missed some plot points. For example, Joe doesn't try to steal 22's badge, she doesn't want it. Later on, 22 doesn't steal Joe's body, she is accidently placed in it. And Joe is supposed to be dead.
I am also confused about your interpretation of the barbershop scene. 22 isn't a better version of Joe because she is "white", but because she is young and curious.
In the end, Joe saves 22, so I am not sure how you interpreted this as being a "white savior" example. They both save each other.