I realize that this is all a matter of personal opinion and that there are passionate views on both sides of the "prepare for any eventuality, vs. travel light" debate. I skew decidedly toward the "travel light" philosophy, and I would suggest that if you're considering bringing a stroller just to haul your stuff around, you're probably bringing too much! Approaching the Disney parks with the packing efficiency, ease and flexibility of one backpacking through Europe is, in my opinion, infinitely preferable to the unwieldy burden of going Oregon Trail-style, with all of your worldly goods in tow. Even my "always be prepared for anything" Eagle Scout DH has learned to leave the overstuffed backpack behind when we set set off for the parks.
We get by with just my cross-body purse (Vera Bradley hipster or smaller), sufficient to hold my sunglasses, camera and phone, a spare credit card, a "First Aid" baggie with a few Tylenol capsules, Band-Aids and blister pads, a 3x5 card with the day's touring plan/FP+/ADRs, a baggie with individually-wrapped hand sanitizing wipes, 4 Dollar Store rain poncho packets if the forecast calls for rain, and sometimes, the kids' little autograph books and pens. DH keeps his phone, a few Kleenex and his rescue inhaler in the pockets of his cargo shorts. The kids keep track of their own sunglasses, which are on cords so they can be effortlessly worn around the neck when not in use.
We don't bring any beverages or food with us: we buy lunch in the park or return to our hotel at lunchtime for an in-room lunch and naps, and you can get iced water for free, on request, at any Disney counter service eatery. If a need for snacks is anticipated, I toss a few granola bars in my purse, which are lightweight and take up almost no space. We don't bring any "diversion" items or electronics with us: we try to plan well enough that we're rarely in line for long, and if there is a wait or long bus ride, we spend it chit-chatting with one another or playing simple, "I Spy"-type games.
Unless you need a cooler to hold a huge picnic meal or to satisfy a medical need (e.g., keeping insulin cold), it's going to be nothing but an annoyance to lug around (as would a stroller -- if you have the freedom to go stroller-free, why shackle yourself to one unnecessarily?). If you need the cooler to keep your lunch cold, then consider renting a locker to house it instead, until it's needed.
