When my twins were younger, I brought my own double stroller. For one thing, it was greatly needed in the airport. I noticed you said you were flying. I can't imagine trying to carry them or even have them walk in the airport. Plus carrying a diaper bag, purse, and whatever carry-on you might have. They did have to get out of it to go through security, but then I pushed them right to the gate. Then it was waiting for me when we got off the plane in Orlando.
It was nice too at the resort. I liked having it when we ate at the resort. I liked having it to push them to the bus stops. Yes, it can be a "hassle" to fold up to take on the busses, but still worth it. My husband took care of folding up the stroller and carrying it onto the bus and I had the twins. We had it planned out. I know that sounds funny, but it was necesssary. This is why....
I think that lots of times other guests get impatient when people with strollers aren't ready. I mean I have stayed at POP three times, and I have seen people see that a bus is coming, and they don't start getting their kids out or start folding it up. Then the bus pulls up and people want to get on the bus and they can't because these people are just starting to get their kid/kids out of the stroller. Then they still have to fold it up. I have seen other guests get very mad at these people holding up the lines to get on busses. I have seen the same thing at the parks too.
We never had trouble with our double stroller and the busses. Yes my husband had to stand and hold it, but it was no problem.
As far as leaving your stroller unattended, I never worried about it. I never left anything valuable in it either. I have only heard once about a stroller getting taken and then it was returned when the person that took it only thought it was theirs because it was the same kind of stroller. I have also heard about people who think their stroller was taken, when it was really only moved by a stroller attendent CM. We tried to put something on the handle to easily find our stroller. A few days we used a balloon from POP.
Now last year I rented a double stroller, and this year I will too, but they will have just turned 5. They can walk to and from bus stops and around the resort. We aren't flying either.