I use shampoo for washing my hands. (I've got a particularly dirty job. (hence the name))
If you could see what it takes off your hands... You'd never put it in your hair! Shampoo causes dandruff. So, Yes, Changing to mouthwash could actually improve things.
Oh.. And there's no other area on your body as white as your scalp. That's pure bleaching of the skin from detergents used in shampoo.. So. Yes, it could even brighten your smile a bit. I suppose