Here is my experience and opinion. I just came back from Disney last week. Spent 10 days down there with the family (children are 6 and soon to be 4). I consider myself an avid amateur photographer who is willing to spend a little extra to have some nice gear.
This is what traveled with me:
Nikon D5100 body
Nikon 70-300mm f/4-5.6 VR -
http://www.nikonusa.com/en/Nikon-Products/Product/Camera-Lenses/AF-S-VR-Zoom-Nikkor-70-300mm-f%2F4.5-5.6G-IF-ED.html
Nikon 16-85mm f/3.5-5.6 VR-
http://www.nikonusa.com/en/Nikon-Products/Product/Camera-Lenses/AF-S-DX-NIKKOR-16-85mm-f%2F3.5-5.6G-ED-VR.html
Sigma 18-35 f/1.8 Art -
http://www.sigmaphoto.com/product/18-35mm-f18-dc-hsm-a
In all honesty, I left the 70-300 in our hotel room. I was able to get 90% of all the shots I wanted with the other 2 lenses. Sure, were there times I wanted the extra reach such as being at Animal Kingdom, yup. Was it worth it to lug it around the park. IMO, no, therefore it stayed in our room.
Did I wish I had a 70-200mm f/2.8 for Fantasmic and a tripod, sure, but I sucked it up and made do with what equipment I did bring. It simply was not worth lugging around a 70-200 f2.8 or 70-300 f/4.
Now with that being said, my plan was this. I would use the Sigma 18-35mm for fireworks, the lighted parade, and character meal/indoor character photo opportunities, and indoor rides (obviously with no flash). I used the 16-85 for everything else. Yes there where times I kept the 18-35 on for some outdoor shots and vica versa.
Some have complained that the Sigma is a bit heavy. In all honesty, it does have some weight. However, that is the price you pay for having a very nice piece of glass. All in all I was very pleased with the photos I took. For a 10 day trip I have about 3700 photos. Yes there are some great shots and yes there are some duds. That is just the way it is. I took about 200 photos of the lighted parade for which I have maybe about 20-40 really good shoots. Same with the Magic Kingdom fireworks. (Full disclosure, we had reservations for the desert buffet and had a very nice unobstructed view of the castle and the show). Sure I wish I had a tripod, but I just jacked up the ISO and try to hold still has best I could. Out of the 100-150 shots I have about 20 that I think are pretty good. (there are some that if you look really hard, it might be just a tad bit fuzzy for the castle, but for the casual observer, they think they are wonderful)
It comes down to your preference. Don't over think what you will take as far as equipment wise. Plan ahead what type of shots you want to get and plan accordingly. Except the fact that you are not going to get every shot you want. At times it simply will not not practical. Finally, know your own abilities and accept them. I'm sure if I took one of my good friends who is a semi-professional, he would get some better quality photos than me. Simply because he has been doing it a lot longer than myself.
Overall I would not change a thing. We had a great trip and I have some wonderful photo memories of my family to last forever.
I have posted a few examples for your reference.