I agree about bringing your own along, the airport, to and from your resort, waiting for the bus, etc. Having a place for a little one saves a lot of whining from child and parent!
I have never heard of anyone having a stroller taken a WDW, but I guess the possiblity is real. My husband used to take a bicycle lock and lock our wheels so that if someone started to take off with it (accidently or not) it wouldn’t move easily. But it could still be dragged to a new location if a CM needed to reposition it.
Your child is going to be doing a lot of riding (when mine did when they were that age), you want them to be as comfy as possible.
Our DD just turned 5 and we have just purched a new stroller just for the next three trips (over the next 10 months). I paid $60 for it, but saved a ton vs. using disney’s strollers. Here’s the one I just bought. It’s made of a water resistant material, has a basket and reclines. Not a bad price!