I did see 4 kids in an Epcot stroller one night..... but 6 would make for quite a picture!!!
I think Disney is just a special case. I saw somewhere that the average family walks between 5 and 9 miles a day at WDW - I don't know if that's really accurate, but it doesn't sound too far off to me! The first day, my 7 year old walked really well. The next day, we went to Epcot and he rode a lot. He still walked a lot of the time during the week, too. We were there for 7 days....if you figure 5 miles a day, that's 35 miles? And if you figure his strides are half as long as mine, it's probably more like an adult taking enough steps to cover 70 miles in 1 week.

We tried to stay well rested, but I tell you, by the end of the week, those scooters for rent were looking mighty appealing to me!
I think that Epcot is the most crucial park to have an extra stroller. We had a Maclaren for the 2 year old that was sometimes used for the 7yo. We did fine on some half days at the MK without an extra, and at MGM, we were fine without an extra. We only did the one day at AK, and did fine without it, but I imagine if we had gone later in the week, another stroller might have seemed very appropriate. It's a long way around that tree o' life sometimes! After the first few days, we would ask my son at the start of each foray into a park if he thought he'd need a stroller for that visit. He made really good calls, I thought.
Glad the stroller question resolved well for the OP!