I'm going to skip over the replies and give my opinion because I can only guess what 6 pages of replies on a stroller thread would mean.
When you said "older kids", I was expecting you to be talking about 8 and 10yo kids. IMO, at least at WDW, 4 and 7yo's, esp a small 7yo, are not older kids as far as walking around the park all day.
I always brought a stroller for me son. He was almost 5yo his last trip and it was nice having the stroller. Mostly so he could sit, but also so we could have somewhere to put our stuff (equally nice for that...

).
I'm taking a trip next week with only him and it will be his first trip without the stroller (he's 6yo now). Since it will be only the two of us we will go at a slower pace, not to mention I wouldn't want to lug a stroller from home.
Maybe on the Epcot day I will get a stroller, but I seriously doubt (I won't even mention the possibility to him). I'd say the only real chance would be if he were to get sick and just not be feeling 100% since he always seems to get a cold when we're there (nothing to keep him in his room, but just enough to sap some energy).
The problem with him and strollers is that then I'm pushing him around all day, exhausting myself, and he's saving his energy for later when I'm ready to collapse.
You should make the decision that works best for your family. BTW, before we had our youngest son, we went to WDW with our then 3 children, and the children were 4, 7, and 10yo. We brought a stroller from home and the 4yo never needed it. I think one time we used it, just because we'd brought it from home and were determined to use it. I think our 7yo would like to have used it, but we weren't going to push him in a stroller when the 4yo was fine to walk.

So it really depends on the kids and how fast you are going to walk and how many hours a day you are going to be out. Also, that was a January trip, so the temps were not hot, which helped their energy level.