I'll admit I was shocked when I read the OP and the article linked to this, but on reflection I suspect we've all carried a sleepy three or four year old very occasionally. Doing it regularly is for me a step too far and I wonder if it is maybe meeting the mother's needs more than the child's. However, no child is going to be harmed by being carried as long as they get plenty of opportunities to run around, so if mum's can cope with the back strain(!!) good luck to them.
I was the first person in my cohort to use a baby sling in 1972 (the little blue one from Mothercare, remember those?) and I got lots of derogatory comments, but I couldn't afford a pram. My daughter went from a carrycot on wheels, to a sling, to a pushchair. Using a sling was lovely, but I think I probably stopped using it by the time my baby was about nine months old. Three years later my second child had a pram (we'd gone up in the world by then) and so I only used the sling for short bus journeys with her.
I know the mums in the article say that their children can walk long distances, but I'll never forget the shock I felt when taking a class of primary school children for a walk to the sea. They were all 7 year olds and the journey from school to the sea is just over a mile, along a beautiful river meadow. We had done about a third of the walk before some of the children were asking to be carried and two were actually in tears because they had never had to walk so far.