What a child can do is not the same as what a parent should allow them to do or what a parent should force them to do.
If one left whole-grain crackers and cereal on the counter and stocked the fridge with milk, cheese cubes, hummus, fruit, and cut-up veggies, a three-year-old could certainly feed herself whenever she was hungry and satisfy every nutritional need. We don't do this because children have emotional needs as well as physical ones. Children need to be nurtured/cared for by human beings for their brains to develop properly.
Children need to know that their parents are taking care of them and will defend them and protect them and keep them safe. Children need to be smiled at and hugged and talked to and responded to. Offering food is part of nurturing. Children learn that they are cared for, and that their parents want them to be healthy and happy and are willing to work -- and argue with them -- to make that happen.
The age a child should be made to take responsibility for feeding themselves varies from child to child and culture to culture. A twenty-two-year old living in your house is old enough; you may negotiate an arrangement in which you provide the meals, but it is not required. A six-year-old should not be made to do this. They still need nurturing from their parents; their brains are still developing. There might be a situation where it is necessary occasionally, or even for 1-2 meals each day -- e.g., if a single parent has to work long hours -- but that is in the context of necessity, and you make it clear that it's not by your choice. You don't do this because you decide it's too much bother to feed your kids when you could be playing video games. (Though letting your kids get their own cereal on Saturday morning, thereby allowing you to sleep in, will not hurt them, and teaches them that they can do things on their own.)
Beyond this... if you are doing a good job as a parent, making your kids feel loved and secure and teaching them how to take care of themselves and to do some chores around the house, at some point they start to feel ready to take on the responsibility of feeding themselves. At this point, encourage it!
Let your five-year old make her own PBJ, let your seven-year-old heat soup in the microwave. Teach your twelve-year-old to make a pot roast for the family dinner. If you are in the kitchen, ready to eat with them when they are done, wonderful; this is part of helping them grow up. If you need to be gone or sleeping when they do this, all right, it's necessary. If it's the weekend/vacation and your kids are independent and comfortable with fixing themselves a nutritious meal for their lunch, you can let that be the plan for the day. But until they are eighteen (on average, and in most U.S. cultures; children mature at different rates, and cultures vary), you should provide at least one sit-down meal daily (home-cooked by preference, but a healthy breakfast of cereal, fruit and yogurt works if it has to), during which you take the time to demonstrate that you are an engaged, caring parent -- no matter how pressed for time you are.