I would not say that the child gets a wider variety of experiences in child labor compared to child care. Child care is explicitly designed to have a variety of activities for children. Child labor is the low skill kind with lots of repetitive actions. Child care also takes the child's needs into account, like having nap times and eating times designed around children, instead of around a production process.

As for caring about the child's well-being? The parents, which is why they should hire the best people they can afford to do various parts of taking care off them. Hire the best doctor for their medical care, hire the best dentists for their dental care, hire the best teachers for their education, hire the best child-care professionals for daycare.