Immediately after he weaned, I got pneumonia and he got sick 2 days later. He couldn't keep anything down and was hospitalized for dehydration. There are other medicinal uses for breastmilk. I've put it on diaper rash, cuts and burns on my kids, and the wounds healed faster. I've collected it and put some in a syringe to put a few drops in a baby's clogged nose - it can ease congestion.
Mom gets benefits from extended breastfeeding as well.
The longer a mom nurses babies in a lifetime, the more she reduces her chances of getting breast or uterine cancer. A minimum of a year of nursing is needed to reap this benefit. Take what you like and leave the rest.