What a load of crap.
I eat a wide-range of food products.
Excuse me, there would be no need to slaughter cows and chickens, as many are bred to produce milk and eggs. So if there was no demand, then none would need to be bred.
Cow's milk is also full on anti-biotics which are not removed during pasterisation.
Also, we drank our mother's milk, as that is what it's produced for, feeding the offspring of an animal. Since we're not cows, why should we drink the milk of another animal?
Secondly, in the production of milk, many calves are killed after the mother has given birth, just so she can produce milk.