Arjuna was a spiritual traveler of the highest order. Lord Krishna imparting wisdom of Bhagavad Gita only to Arjuna means Arjuna finally succeeding in reaching state of enlightenment in the battlefield. The wisdom of Bhagavad Gita can be understood only by one who successfully reaches state of enlightenment and never anyone else.
Amongst all the characters in the battle of Mahabharata Arjuna was the one and only one competent to fathom jnana wisdom of Bhagavad Gita.
Even in ordinary life not every seeker, traveler of spiritual path succeeds reaching state of enlightenment in ones lifetime. Reaching God in one’s lifetime was not that easy. One needs becoming a Swami Vivekananda in real life to reach God, reach our true inner self existing in heart. Anything less than Swami Vivekananda would not do.
If we feel we possessed the mettle and grit exhibited by Swami Vivekananda, we can fairly expect to succeed reaching state of enlightenment followed by moksha salvation.
Allegorically Lord Krishna represents our true inner self, soul atman existing in our heart. What actually happened in Mahabharata was, in the battlefield facing the enemy head-on Arjuna suddenly reached state of enlightenment and in those moments he realized he was primarily a spiritual traveler and not the manifest physical human form facing the enemy.
Immediately his attitude changed from cowardice to fighting the enemy head-on. For the sake of his countrymen and soldiers he finally prepared himself to fight the enemy consisting of his cousins, close relatives and friends and also maternal grandfather Bhishma Pitamah and Guru Dronacharya. In those moments as an enlightened being Arjuna overpowered death forever.