There are two ways in which animal’s DNA is genetically modified: 1) in labs, and 2) through breeding. We do this so that the animals are better suited for our purposes than for their own purposes.
Genetic modification has been a fact of animal farming practices since domestication of livestock animals began some 12,000 years ago. Animals are selectively bred to achieve specialized breeds for meat, milk production, egg production, fur, leather, wool, down, and other animal products. The domesticated livestock animals we have become so familiar with today through millenia of genetic modification via selective breeding have been rendered unfit to survive outside of captivity. In many cases such as with pigs, turkeys, and “broiler” chickens, animals have been bred for size to such an extent that when allowed to live the rest of their lives past slaughter age (45 days for “broilers”, 4 months for pigs) they grow so large that they develop chronic health conditions which they ultimately die from prematurely. Pigs that have been rescued from farms that are left to live out their lives in peace often have their hooves split in half because of the extraordinary amount of weight being applied to their feet because of their unnatural size (made possible by genetic modification to suit our needs for fast meat, rather than their own needs for a long and healthy life). Turkeys who outlive their slaughter weight/age in sanctuaries often have to undergo surgeries on their uterus because when they lay eggs their uterus is pushed out.
None of these are valid reasons to continue eating and exploiting animals. We created their suffering; perpetuating it does not justify our actions against their genetics. The answer is to eliminate demand, which will eliminate supply. This means that we will no longer be breeding livestock animals for human consumption.