I would go with the stroller at the airport. Just make sure if you take it to the plane, that they gate check it properly. Once I did this, and for some inexplicable reason, they took the stroller from me at the gate in Dallas, and put it in the belly of the plane. When I arrived at Dulles, I had to walk through the airport, up and down stairs (non-working elevators) about a mile or more carrying a sleeping toddler, my purse, my carry-on and diaper bag. There were no skycaps or anyone to assist with mobility. It was a nightmare!! I had to pick up my stroller on the baggage carousel. Now, if you don't know Washington Dulles, the baggage carousel is a good mile from the arrivals terminal, in a separate terminal on a different floor. I was livid!! It was United Airlines that did that to me! But anyway...... it is my preference to push the stroller and hang my purse or the diaper bag or anything I can from the handles or under it. If your child would rather walk, and is moving in a forwardly direction, you can always just plop your carry-on's into the stroller and let you child walk along side. When moving through a crowded airport, we always liked those "hand holders". The tether things that non-parents call leashes. Now that my DD is 8, she uses the "hand holder" to attach to herself and her doll or stuff toy.
The rental stroller is perfect for older kids like mine who really can't justify using a stroller in normal public. When Disney is hot and humid and crowded, even an 8 year old needs a stroller. Luckily mine is petite and fits in one. The big advantage of your own stroller is that it probably has that basket underneath. The Disney ones don't. I really like that for purchases.