I've been twice in the past 3 years with a stroller both times. From my feeling the theft concern is overblown honestly, it has happened, but I think it gets a lot of attention when it does compared to the number of strollers.
1) you can get those GPS tracker things if you really want to recover it.
2) don't leave anything of value in the stroller. We generally have left our backpack in the bottom and the cooler, but realistically its just a change of clothes for the baby, diapers, some snacks and water.
3) Disney is the Indy 500 of your stroller life, its what you've been training for, its an intense workout of both you and the stroller... you want good equipment. I'm jesting, but seriously, it is more miles than you put on your stroller regularly, you're pushing and pushing and it requires living out of it for each day, all day. I wouldn't want to have an cheap umbrella for that. Why have something decent if you don't use it when you need it. You're buying a cheap old car to keep the miles off of your new Mercedes, but not enjoying your new Mercedes.
4) I think the whole concept of renting a stroller for a week at over $100 is a waste of money. Use what you got. or if you must buy something on craigslist. Our strollers are more in that 150-300 range so maybe it doesn't directly apply to the $1K threshold. However, if you rent twice, you basically paid for a stroller, which is why I take the risk.
First time we went we bought a Graco stroller on craigslist for $50 to not take our Britex stroller and keep it nice. We ended up using it and liking it so much we eventually sold the Britex that we kept so nice and running the Graco into the ground over a number of years. 2nd time we took our Joovy Caboose.