This view is based on a misunderstanding of what moral responsibility is. Moral responsibility is essentially credit assignment [1]. It exists as a concept not because it matters whether or not you actually could have done differently before, but rather as a mechanism to teach you to do differently in a similar situation in the future. "You" is just a machine making decisions. When you make a bad decision, according to the other machines surrounding you (society) or your own brain's reward system, you are assigned blame or feel guilty because you should avoid making the same decision again in a similar situation.
To say that moral responsibility requires something supernatural doesn't make sense, since we successfully use the concept all the time. Its a mechanism for multi-agent learning.