We have 3 strollers (a high priced one (Peg Perego), a medium priced one (Graco) that came with our travel system and a cheap umbrella one (Jeep)).
We bring the medium priced one (Graco) to all of the parks we visit (DL, LegoLand, Sea World, Zoo, etc) since the umbrella stroller is too small (and I don't want to carry the backpack or diaper bag the whole time since it doesn't have a basket) and has a crappy sun shade.
The medium one reclines, has a great sunshade, cup holders, etc and a HUGE basket.
I've heard of expensive strollers getting stolen, so yeah, I wouldn't bring anything you would freak out about if it was stolen. I don't think it happens much, but better to be safe than sorry.
I'm a big germaphobe (LOL), so renting one just kinda grosses me out. Never know who might have taken a poop and didn't wash their hands good and then pushed the stroller....and then you push it, eat cotton candy and lick your fingers afterwards (ewwwww) LOL
Yeah, you get the point.