We just had a trip like this. I have a back/neck/leg injury, and had trouble standing for long periods. The kids were "ballast" when I was pushing (my dd gets very stressed in crowds). My mother used it because her knees hurt from all the walking. My Dad sat in it sometimes (he still walks with a limp from being hit by a car a couple years ago, but is too stubborn to use his own assistance device).
Yes, we traded off. Frequently. I could stand for a little, and sometimes needed to stand to stretch the muscles in my back and change the tension on my nerves. Yes, we joked about it being the best ride in Disney world. Yes, we joked about it getting us in first everywhere (it didn't). Yes, we joked about always having a place to sit. Yes, we will do it again, even when I don't have a back injury, because my parents and child benefited so much from a place to sit when their bodies had had enough.
We wouldn't call it a lawn chair on wheels, but we did call it all manner of other things. I joked around, trying to push myself fast enough that my family couldn't catch up to start pushing. We'd say "Hey, wheelchair-lady! I'll push you! This is where you wanted to go, right?" We played "run-away wheelchair" when there weren't people in the way.
And yet, when I tried to go without it? I had to take percocet and muscle relaxants just to be in little enough pain I could move. Appearance isn't always reality.