Answer
No message in the Shrimad Bhagavad Gita suggests that the purpose of any of our actions should be to prove ourselves 'the best.'
Yes, certain special actions — yajna (here, the reference is not to fire rituals; one should refer to the various forms of yajna described in the fourth chapter of the Gita), tapas (austerity), and dana (charity) — are stated to have the purpose of purifying oneself, as the Lord says in this verse:
"Yajno danam tapashchaiva pavanani manishinam." (Shrimad Bhagavad Gita 18.5)
The second purpose of action is loka-sangraha (welfare of the world), as the Lord clearly states in this verse:
"Karmannaiva hi samsiddhimasthita Janakadayah | Lokasangrahamevapi sampashyan kartumarhasi ||" (Shrimad Bhagavad Gita 3.20)
Loka-sangraha means the welfare of the world — keeping society united and maintaining its proper order.
The Gita also teaches that action should be free from selfish desire, free from ego, and performed as an offering to God, for God's sake:
"Mayi sarvani karmani sannyasyadhyatmachetasa | Nirashirnirmamo bhutva yudhyasva vigatajvarah ||" (Shrimad Bhagavad Gita 3.30)
To act with the feeling 'I am the greatest' is clearly a matter of ego.
As for the idea of drawing a longer line instead of erasing the shorter one — one must also bear in mind that sometimes it becomes necessary to erase a line, whether it is long or short.
The war of Mahabharata was fought precisely to erase the line of injustice and draw a new line aligned with dharma. That line deserved to be erased.