Cows' milk has to be used, or they're uncomfortable. And guess what? In nature, humans wouldn't milk them. Who else do cows produce milk for? Their babies!
That's right: cows' babies are supposed to drink their milk. In most scenarios, cows' babies are taken away and either killed and disposed of, turned into veal, or grown up without their parents to produce more milk. The males are always killed or turned into food.
Chickens is the same thing -- at the Calgary zoo they have piles of dead baby chickens. After they're born, the male chicks are all killed and given to the zoo to feed the birds.
So yes, if cows aren't milked it's uncomfortable, but the milk is supposed to go to their babies, not to humans.