I also use the food processor to shred cabbage. Before I got the enormous food processor, I would stack a few cabbage leaves, roll them into a tube, then feed the tube under the knife that my other hand was rocking up and down. I cannot imagine using a vegetable peeler to shred cabbage.

I also just wash the outside of the head of cabbage. It's pretty tightly wrapped, right? And doesn't grow with leaves partly underground like leeks.