I don't think anyone doubted the story. The details might be questionable, but the basics that he tried to fight with elephants is highly likely. We have plenty of sources for War Elephants in his time, so the idea that he didn't have them would be the larger surprise if someone could prove that.

I think it would be more accurate to say "based upon true events".

It's more like, the surviving written histories and the archaeological record are each giving us only a part of the real truth. It's as if they're both grasping in the dark at two different parts of the same elephant.