Chick starter should be at least 20% protein to support their rapid growth and feather development. The reason adults need extra protein during molts is because layer feed is usually about 15-16% protein. It's just barely enough to support egg laying, but not quite enough for a molting bird.
You don't need to give them more protein, but it won't do any harm either. One of the first treats I give my chicks is live mealworms.