I feel like by saying people do things in both racial categories, you actually missed the entire point of fighting against racism.
Fighting against racism doesn't mean that Black people never do anything wrong. It is an acknowledgment that white people, in this case white women use their privilege to disrepsect Black people in a way they cannot respond to in kind.
Imagine a senator cursing you out or hurting you. Because of their power proximity, you would not be able to respond freely. In this same way, Black people have to deal with white people yelling at them and abusing them. America has a long history of lynching Black people without any trial. This is not a two way street.
I feel that you had good intent when you read the piece and wrote your answer. I appreciate that. But I would respectfully ask you to dig deeper on this issue.