"Go extinct" doesn't mean "be all killed" or "be violently sterilized". It means that we should stop breeding these unnatural and disadvantaged animals. The white tigers that exist would live comfortable lives in captivity (because they can't survive in the wild) and die naturally of age. Everything about their lives is already artificial, but animals don't seem to need or enjoy sex in the general sense in a way that humans do (they have no interest when not in heat) and therefore it's not inhumane to prevent them from breeding.

For a more severe and human example, people who have a high likelihood of carrying the recessive Taysach's gene are often tested. If they both are carriers, they adopt, rather than taking a 1-in-4 chance of having a child who will die horribly. If one is a carrier and the other is not, then it's no problem. No one is being sterilized; it's positive, voluntary eugenics and it's a good thing.