The email would be helpful if the person you are giving the feedback to wanted to ask you a follow up question. Also, it'd provide a way for you to track all the times you've given anonymous feedback or asked for feedback in one place.
What if I want to give feedback, but don't want to hear a followup question? Aren't you interested in this kind of feedback? And if I don't care about tracking?
Plenty (if not the most) of first time users don't care about these things, you are losing them.
