We've done it all on separate trips - rented a house, stayed offsite within walking distance, stayed offsite with a hotel shuttle, and stayed onsite. The time we rented a house, we were with the grandparents, and we just thought everyone should be comfortable. It was really great, but we have little kids and only went back to the park half the time after we came back for the afternoon nap (sleep is very important for everyone's mood in our family!). We did make sure we got there at open every day and really felt like we got a lot done. Plus, we obviously go a lot so we weren't concerned with seeing everything. Since your kids are older, I'm sure you'd be much more likely to return every night.
The things I liked about staying offsite included:
Having 2 full bathrooms
Lots of space to run around including a sizeable backyard
A bbq for cooking dinner as well as a fully stocked kitchen (we cooked dinner pretty much every night)
Lots of games and dvds available for the nights we didn't go back to the park
Didn't like:
Driving back and forth from the park - very time consuming compared to just walking over
I found the rental on vrbo.com. If you're interested, I can PM you the listing page. It was a reliable 10 minutes to the Mickey & friends parking structure driving on surface streets (no traffic concerns to speak of) and was really clean (a thing for me) and nice.