Farm animals would go extinct
This argument is dishonest and assumes that a life of captivity, mutilation and selective breeding for the purpose of exploitation and slaughter is beneficial to the animal, that we are doing them a favour by forcing their existence.
Not wanting farmed animals to go extinct is a selfish concern, not a concern for the animals which are never brought into this world to enjoy it, to live a natural life as nature intended. Farmed animals are selectively bred and modified by humans to be profitable. They suffer all kinds of health problems because they are bred to be much bigger, faster growing or high-yielding compared to their natural ancestors.
It is true that they may go extinct, continuing to breed farm animals serves no logical purpose when there is no demand for their exploitation. Is this a bad thing?
The world we have created since the dawn of animal agriculture is not one of nature, it may have served us well in the past when the human population was low and animal agriculture relatively small. But we now live in a world of eight billion people with consumption demands that far outstrip what animal agriculture or the planet can provide.
Humans and livestock now make up almost 96% of all mammal biomass on earth. All other mammals account for just 4%.
But total extinction is not a realistic future. If we really wanted to keep the pigs, chickens, cows and other animals we eat alive, we can conserve them in the same way endangered species are, by not by killing them. Many animal sanctuaries exist today where farmed animals are rescued and enjoy the rest of their lives, there is no need for species to go extinct.
Read more on this topic at the Ethical Vegan's website.