If you want to suggest that 16 is a great age to allow people to start work, I don't think you're going to find much resistance. That's true in the United States, and much of the world, as you remarked. But 13 is nowhere close to 16, so that's where you're seeing resistance here.

Another point that I thought it was obvious, but perhaps it isn't, is how easy it is for older coworkers and bosses to manipulate children. Kids don't have the experience, and they don't have the experience or composure necessary to reliably walk away from bad work environments. So then, is there some totally necessary societal function that we desperately need young teenagers to feel? If there's not, why don't we take the risk off of them.

And finally we have to come back to the elephant in the room. In reality, people who propose allowing children to work are doing so because they don't want to pay adults more.

And again I think it's obvious, but maybe it's not so obvious to others, that if the goal is to give kids a variety of experiences then there are plenty of great ways to do so. Sports, music, school, volunteering, extracurriculars, you name it. Structured environments with proper supervision, managed by people who care about the safety of those kids, and aren't going to try to make a buck by mistreating them.