I certainly understand what others have said above and I made some assumptions that seem to be true- mainly that resupplies are frequent and near towns so items like fresh food, local health care, new diapers, etc. will be easy to reach. I assume this is not the PCT.
The carrier point is a good one- a full back carrier will likely not work for the child. However, many front Bjorn-style carriers might be great. There are many parents in the "attachment parenting" camp who cruise their daily lives with kids following their schedules. I'm not of this group necessarily, but it's a common mindset. I do know that my son had some of his best and most consistent sleep when traveling in a carrier.
Last, I don't think this is such a horrible and selfish idea. Sure- taking my son backpacking at a very early age was partly my own self interests at play, but these trips are things he loves and talking about backpacking has been a major part of his first words, conversations, and memories. The trip proposed is of a different scale, sure, but done correctly and thoughtfully, this could be a wonderful experience for the child.
And if the trip is as I imagine it, the parents could call it at many, many points on the trail if they need to.
Last, thank you Roger for your points about raising children. They have helped to cement my perspective on raising my child.
P.S. Isn't it interesting that we're all talking and the parents are not here at all. In all likelihood, we only talking theory.