Are push-ups bad for a 10-year-old boy?
Yes, it is definitely safe. Kids do things that we normally wouldn't think of as being potentially harmful to their joints or muscles, such as lugging around heavy bags of raked leaves, carrying out the garbage, picking each other up during play, carrying around dogs or their toddler siblings, bowling, and carrying heavy backpacks on their backs at school.
Push ups, on the other hand, stress the child's body equally on both sides, with very minimal strain on the back. If your child wants to train at push ups, encourage it. It won't do any harm or won't cause any stoppage in the growth of your child.