I can confirm this. We took our little one, and EVERYONE offered to help me fold our stroller.
We brought a GB Pocket, which was very compact when it folded. And it's made open and close FAST.! Since then, there are more models that do this, and are better for younger kids. It didn't have a big basket, but I found a bag that would go over the handles to hold snacks and maps.
You didn't mention this, but don't let anyone discourage you from bringing a toddler. Children under 3 get in free, you can feed them at the buffets for free, and they REALLY believe in the magic. I wish I had pictures of my son from the Disney Jr show. He absolutely adored his time at Disney, which made our time there more memorable.