This is why it is unconditional. He is responsible for it. It's not like the Old Mosaic Covenant where man had to do his part in order to be in accepted by God. In the New Covenant God is 100% responsible.
He elects those who are His and nothing can snatch them from His hand.
Actually, the verse is "no man is able to pluck them out of my Father's hand." And no one can pluck us out of His hand. We can, however, choose to reject God.
Your interpretation goes to one extreme and, again, leaves us with no responsibility at all. However we get to this point, we can know that somewhere we made a wrong turn.
Well no matter how you view it, it is conditional. It is conditional on God putting it there. Although there is no eartly condition upon which grace is bestowed, it is still necessary that God grant that grace as a condition to Salvation. I believe the proper term rather than "unconditional" would be "unmerited". God is not arbitrary. He has His reasons. We just don't know what they are.