That's a funny show. I watched on the weekend, when I noticed that my Tivo recorded it. (It records any show that has Disney World in the description)
Kate cracks me up, mostly because I would have reacted exactly the same way. I only have one son but he was never allowed to have Ice Cream cones, Cotton Candy or any thing that had major Sticky Mess potential when we went places when he was little.
At the age of the 3, I would have given him the Ice Cream only if we were sitting at a table or if I was holding it and spooning it into him. Just like Kate.
When I see a little kid with an ice cream covered face and shirt,

I shudder.
My mother always thought I was over-reacted about this too.. but I never had to deal with a gross sticky child when I was out and about.