We used a cheap umbrella stroller in WDW when SDD was 7. There was no way she could have kept up the same pace as the 5 adults she was with. She mostly used it when we were crossing long distances, getting from park to parking lot, and she would sleep in it on our way out of the park after the fireworks. It was also nice to have a place to hang our lunch bag.
I know this is a bit off topic but can you imagine sitting in a stroller and being pushed around 

You walk and stand so much at the parks. I think it would be really hard for a 4 year old to be expected to do that without you carrying them a lot.