I think Disney is just different. My older kids stopped using strollers basically by age 3.5, but continued to do so at Disney. On our last trip, we had a double stroller and umbrella stroller for a 7, 5 and 1 year old. It's just too much walking and, frankly, is more convenient for us adults. I really think of it as more of a convenience for my wife and me than for the kids. We can move through the park much quicker with a stroller than we would if we had to cajole the kids along. Our older kids could definitely do it (and are headstrong that they will), but they would complain at some point and be happy for the ride. Most importantly, not using a stroller for them would slow us down. We are single-focused in the park and try to move quickly from one attraction to the next. The strollers help facilitate a more efficient day. Plus, they are nice to hold bags and other items when we hit the rides as the other posters have noted. That also means, no backpacks in the sun!
That said, next fall, we will be going with an 8, 6, 3 and 1 year old. We plan on getting one double stroller and one umbrella stroller for the lot of them. We just couldn't talk ourselves into two doubles. That may mean one of them ends up on my shoulders from time to time, but so be it. The 8 and 6 year old will take turns walking and if they are too slow, I am fine putting one of the four on my shoulders if I have to.