I did a
blog post about strollers! When you say that you're staying at the California Spa and Resort, do you mean the Grand Californian? If so, it's an extremely easy walk from the parks.
As other people have pointed out, the primary disadvantage is that you cannot take it outside of the bag check, so not back to your room or hotel or Downtown Disney at all. Depending on where you're staying, this could be a huge problem. If you're actually staying in the Grand Californian it's not as big of a deal.
We travel quite frequently and find it much easier to actually use the stroller in the airport and check it at the gate. However, we discovered last month that American Airlines has changed their stroller policy!
Their official policy is here, but the gist of it is that if the stroller is considered bulky or weighs more than 20 pounds, it has to be checked at baggage with your suitcases and not at the gate. We have a City Mini (which is actually the same as the rental strollers at DL except the DL ones are stripped down). The enforcement of this policy seems to be hit and miss. LAX didn't care at all, Dallas/Ft. Worth mentioned there was a new policy but didn't say anything beyond that, Reagan National we actually had to physically remove DS, fold the stroller, and have them weigh it (18 lbs) before they would let us take it. Chicago (ORD) were also sticklers about the policy. We were told at Reagan National that ALL jogging strollers or 3-wheeled strollers and double strollers were forbidden, though the policy doesn't specifically say that. The policy says "large" strollers and that seems to be a matter of interpretation. It was a real hassle for sure, so keep that in mind if you're flying American. We have never had that problem or heard of that problem on any other airline though. The good news, I guess, is that if you check it at baggage they don't make you pay for it. And any airline will let you check the stroller at check-in if you don't want to carry it all the way through the airport.
Tons of people just buy a cheaper one at Target (which is actually a few blocks
south of DLR on Harbor

). If you have a car, that's not a terrible option. And tons of people abandon them in DTD when they leave. No joke--when I worked there and the store closed at Midnight, we'd get out and about 2:00am and very frequently see these sad little abandoned umbrella strollers

. Oh, and one time there was an Igloo cooler with a note that said "PLEASE TAKE ME"

.
But I digress. Personally, I really like having my own stroller. I know how to steer it, I know my DS will be comfortable in it, and we don't have much of a problem carrying it around. It's been pretty much everywhere. If you don't want to carry it in the airport, just check it as baggage and get it on your way out.