It’s not even necessarily the same processing plant as where the adult chickens are slaughtered for meat.
While this may be true: I have worked at 4 different chicken processing places (though 3 of them were owned by Foster Farms) and they all handled every part of production from farm to package. Both chicks and adults.
And it sucks being on any part of the line that is dealing with the live animals. It’s one of the reasons I don’t eat a lot of chicken any more. One of my first jobs ever as a teen was cutting the toes off the birds so they wouldn’t be able to scratch each other up if they started fighting.