We have been to WDW many many times since ds was born and always bring our own stroller and it's never been stolen. Not that it
never happens, or can't happen....but I do think it's fairly rare. We always put an ID tag on our stroller, and sometimes wrap a bright colored bandana around the handle so it's easier to spot in a sea of strollers.
Why do you not want to bring the travel system stroller? Because I actually recommend bringing a stroller that will be very comfortable for a 1 yr old. Plus you'll have all that baby "stuff" you need when you're out and about with a LO and the stroller you have probably has a nice basket to put that in. When my ds was that age, I loved having the large basket area, the cupholders for us, and that the stroller had a nice big canopy and was comfortable enough for my ds to nap in (and he would - we almost never went back to the hotel during the day because he napped perfectly fine in the stroller - so that was when we'd take turns hitting the "big" rides).
On our last trip, we brought a big stroller (we call it ds' "chariot" lol). We have several reasons for choosing this over the smaller, more compact one we have. Yes it's more of a pain on busses and it takes up more room in the car (but we fly, so we just have to get it to and from the airport anyway). But it's more comfy for ds to sit in, and more importantly, it's easier for dh and I to push (we are both on the taller side so a lot of those little umbrella strollers are just too short for us). DH usually takes charge of it on the busses and I've even done it a few times and found it wasn't too big of a deal, especially since most of the busses are newer and have nice wide doorways and are basically one or two steps up from the curb to the inside of the bus so it's not that bad.
As far as more compact strollers go, there are probably several out there that would be OK for a 1yr old and fold up smaller than the one you have. How much do you want to spend? The Maclaren strollers (thinking Triumph or Quest strollers) get rave reviews around here but can a little on the spendy side. You may be able to find one in last years model that is less expensive online though.
hth