Do you have any facts to back up the bolded? The last time we called AAA (flat tire on the interstate and our spare was flat), it was at least 3 hours (maybe 3 1/2) before someone showed up. We never got calls updating us (I had to keep calling back to find out what was taking so long).
I've had the opposite experiences.
Flat tire on the interstate (I-4). 30 minutes response. Updated with a call just before arrival.
I've never had to wait more than an hour for AAA.
I just carry basic. But it's been enough each time I've needed it. I pay for a basic membership for family as well, so we've used it many times. My daughter once had a Jeep that had a key problem - it's now a recall on those jeeps - we had six calls in a very short time all requiring a tow truck. AAA does 4 tows a year, but I had a card on each person so no problem.
If you had to wait 3 and 1/2 hours, I can see why you wouldn't be a fan, but my experience has been different.
I've also gotten a few AAA discounts. A few dollars here and there. All adds up. You can use AAA to get a food discount at a couple of Disney Springs spots as well - I think it is 10%, but I usually check all my eligible cards for the best one. Try Earl of Sandwich, WP's that I can remember of the top of my head.
To the OP - I'd probably want her to have the car.
If it is kept up well, I wouldn't worry about your drive with her to Disney. As others have said, change oil, have good tires and a good spare, good battery - not one that is on its last leg, check the belts, brakes and fluids and carry some sort of roadside assistance card - I prefer AAA because I know there is someone out there monitoring the responding truck as opposed to farmed out to various tow trucks by some insurance companies.