Arabs aren't the only horses that can run distance. They are just the most common.
Any horse with a decent build, good conformation, heart, a love of running, and condition can do CT or endurance. No heavy draft horses, because they can't cool themselves well and simply aren't designed for distance. Horses with bad conformation will eventually go lame. A horse without heart will give up. A horse without condition will kill itself.
Arabs tend to fit the requirements very well, but they aren't the only ones. I've met successful distance mules, QHs, Paints, TBs, gaited horses (pasos, iceys, SSH, TWH, fox trotters...), Appys, Standies, Morgans... The list goes on.
I'm currently going CT (and eventually endurance) with a SSH. She is a mixture of TWH, saddlebred and arab (NSH), racker, and grade spotted horses of unknown origin. And she does great.
I urge you to try CT with Giget. If she does well and you start winning, do a limited distance endurance ride. It's hard work, yes. Conditioning is fierce -- "racehorse fit" is a standard. But it's very possible.