Our extended family went in 2007 with our then 5 year old and 4 year old niece.
We rented, which was fine in the parks.
Good thing- didn't have to haul it on/off the disney transportation, which requires collapsing and popping back up every time you get on and off. Also, disney trans. can sometimes be very crowded, so it's in the way and cumbersome.
Didn't have to worry about the snack tray because the kids sat at regular seats while dining.
Yes, the seats are hard plastic, but believe me a tired kid can sleep anywhere in any position. We have hilarious pictures to proove it!
Bad thing - getting from the parking lot or disney trans. drop off to the gate, in the gate, and to the rental counter can be a long walk. This is not bad in the morning when everyone is fresh. But when you've been there all day (or even part of a day) and you have to carry a sleeping or very sleepy 5 and 4 year old back to transportation when you are tired yourself, not good!
Yes, all strollers are parked in stroller parking with no security, etc. Didn't find that to be a problem. Didn't see anyone getting stuff taken or stroller taken.
I like one of the responders responses, take an umbrella stroller to get to/from the rental counter then rent theirs inside. That sounds like the best plan yet!
I have rented strollers at world and land parks and found both to be acceptable and in good condition. Not a lot of storage, though. Keep that in mind if you have a little enough one that requires diapers, etc.