I have never heard of anyone having a stroller stolen at Disney. When we took our ds to Disney for the first time we drove so we took both our strollers. One was a travel stroller and the other was a luxury stroller. They were gifts from my mother and they were extremely expensive. Even though the larger convertible stroller was more to fold up and carry we wanted our ds to be comfortable. The idea that someone would steal his stroller just never occurred to me. I think you should take your good stroller as a a rental stroller just seems yucky to me. Lock the wheels if it makes you feel more secure but your child should feel safe and secure in his/her stroller with all the safe textures and smells.
You will also find it much easier to maneuver a well built stroller through the parks. I am sure your homeowner's insurance will cover the cost of the stroller if it is stolen. How anyone could think of stealing a stroller is incomprehensible to me.......very bad Karma for their child. That's like stealing someone's prosthetic limb.
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