I don’t think this has much to do with the mom “letting” him eat lots of junk food.
I’ve run across a case of this myself, and while it was nowhere near as extreme as this, it wasn’t the kids always demanding junk food. It was the mom who was worried they weren’t getting enough to eat if they didn’t eat all the time. The younger sib was 8 mos old when I met him, and he was already 35lbs. She did things like giving her very overweight 5 year old adult sized portions at dinner and not letting him get up until he finished what was on his plate. Snacks were given out all the time, and they were huge–she complained that snacks at school were too small (typically one or two cookies with a glass of juice or water–just right, it seemed to me). She had been told by the kids’ Dr. that she needed to change the way she fed them, but she blew him off.
Before I met these kids and their mom, I would have said that it wasn’t possible to force kids to eat more than they wanted. But that’s exactly what she was doing.
I would venture to guess that this case is similar–not a case of the kid being “allowed” to eat junk food, but a case where the kid has been forced from infancy to eat more than he actually wanted, and by now it’s a habit. The mom I knew had other issues (she nearly inspired a few pit threads, in fact), and we eventually stopped playing with them, but she was well educated and intelligent. Her kids’ Dr. and other moms at the school tried several times to talk to her about it, but she refused to listen. Her problems had nothing to do with a lack of knowledge about nutrition.
Based on that experience of mine, I’d say the mom here needs more than just education–I think a bit of counseling would likely go a long way.