I really do appreciate what you are saying, but MANY of us have been through this same exact situation for decades, without wagons. We all used strollers and made it work.
There are so many great strollers for this available now that I would have loved to have had when my kids were younger. There are strollers built for larger kids with special needs and many on here have used them. That really is the best option for safety for everyone.
Disney banned wagons because they are dangerous to other guests and to the kids riding in them. They are low and can not be seen in the large crowds. Guests are often tripped by them (personal experience), sometimes falling and possibly falling in to the wagon.
Have some folks been given special permission on arrival, yes. But the reports are that there has to be a strong reason for the few that are handed out. Things like PP noted medical equipment needed, physical disability a stroller can not work for. But most reports I have seen are they are not successful in being able to use them. Also they are not giving the red stroller as a wheelchair tags for them. My guess is that guests were trying to switch them out after day one. The wagons now get a special tag in another color, saw them a few weeks ago.
In the end you could waste lots of time if you arrive at security or gate with a wagon. You might have to take it all the way back. I've seen several being told in parking lot go ahead and return it to your car and not be given the chance to take it inside or ask. It is completely up to the CMs at the parks.