Would you still wish that if someone posted racist comments using a fake account under your name, with your picture?
Even a true picture can give rise to cases of mistaken identity. Someone who looks a lot like you could appear on a billboard, and people who see you out in public assume it's you from your face.
That isn't logically inconsistent, i.e. "if you don't want things associated with your name forever, don't put them on the internet under your name". If someone posts racist bile under a pseudonym, it also won't be associated with his or her name.
kazinator's comment is the real issue with that idea, I think.
Even a true picture can give rise to cases of mistaken identity. Someone who looks a lot like you could appear on a billboard, and people who see you out in public assume it's you from your face.