Truthfully, I don't understand why folks have such an aversion to getting a full sized horse for their children. I started on a miniature donkey (who was, by the way, the first critter to ever buck me off and first to ever kick me) but graduated up to a 16hh quarter horse who was a perfect gentleman 100% of the time. Standard sized horses are more plentiful so they are easier to find, they have a better chance of having been trained by an adult who knew what they were doing, and they are just as suitable (if not moreso) for children because they, in general, have a better and more willing attitude than most ponies.
The way I see it, if the kid has a full sized horse, there is no chance they will outgrow it in 2 or 5 or even 10 years. Plus, if the horse begins to have some minor issues from being ridden by children, pretty much any knowledgeable adult (regardless of their body size) can just hop on and work on the issues to keep the horse tuned up the way it should be.
Not so with ponies where the person to tune them up would need to be pretty darn small.