Stopping the war was never the intention, that would betray the fundamental lessons trying to be taught. In context of the story, the idea is that they still have free will but it was a lesson to be learnt for both sides.
If there was a superhuman in real life who could avert all wars (literal and metaphorical), that would be great, but we don't live in a world where we can pray to fix away our problems (my humble opinion of course, not to offend anyone). This sets a basis for the story upon which life lessons can be learnt in the form of the Gita; fundamentally stating that time has no beginning or end, therefore everything on the battlefield has come and gone, regardless of your input in life (" I am death, the mighty destroyer of the world, out to destroy. Even without your participation all the warriors standing arrayed in the opposing armies shall cease to exist.").
I think it's why the book is so powerful, looking at a wiki list of people it's influenced, a common theme is a day to day reference of spirituality more than actual religion [0]
You're right, I didn't mean to imply it's the only philosophy or religion but I would posit it's the largest. Also, the Matrix thing was just a flippant offshoot but I never noticed until I heard them playing the Sanksrit shokla :D
Agreed, Hinduism is a catch all, often misunderstood. It's completely different from Abrahamic religions from a fundamental point of view. You can be any religion, or atheist, and still be a Hindu, in essence. Most (if not all?) branches of Hinduism considers God to be the ultimate reality of which we're all part of as opposed to an external entity, hence in the Gita, Barbarika said Krishna was the one who one the battle despite not taking part directly. Whether you use the Bible, or Vedic scriptures, to reach Moksha (Englightenment) isn't prescribed, it's your journey.
[0] https://en.wikipedia.org/wiki/Influence_of_Bhagavad_Gita