GDP records market transactions. So in your scenario, purchasing the chicken, the chicken feed, the chicken coop, etc would all be measured. The time spent raising the chicken (assuming you did not pay someone to do it), would not be captured.

However, I think part of your criticism is a moralistic and health based argument that raising a chicken is intrinsically better and healthier that Doordashing fast food. When a consumer chooses a $60 delivered meal over a $45 home-cooked alternative, that reveals a preference for whatever they value, such as convenience, time saved, taste, service quality, etc. GDP doesn’t evaluate whether that preference is “good”; it only records the transaction value that arises from it.

So I agree GDP is not a perfect measure, but it's far from useless.