I think sin is to reject God, or be apart from God. A&E rejected God, which is why also we are born apart from God. Jesus came to restore the connection. In that way he can be seen as "atonement" for original sin. But I don't think it is because of his death, it is because of his words and what change they can make in person.
I do not believe there was literally an original sin, since I believe that the story of A & E was allegorical. In the story, eating the fruit symbolizes rejecting God, which was the sin they committed. Jesus came to restore our connection to God.
I agree that it was not so much His death but rather it was His words, since those are what changes a person.
I believe that Jesus came to save us from attachment to the material world, which bars us from eternal life, which I believe is nearness to God.
John 12:24-26 Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit. He that loveth his life shall lose it; and he that hateth his life in this world shall keep it unto life eternal. If any man serve me, let him follow me; and where I am, there shall also my servant be: if any man serve me, him will my Father honour.
Jesus freed us from the chains of bondage, which is attachment to the material world. He gave us His teachings (profusion of His bounties) and then later died on the cross (suffered the greatest martyrdom) so we could be free of sin and attain everlasting life.
“…those who turned toward the Word of God and received the profusion of His bounties—were
saved from this attachment and sin, obtained everlasting life, were delivered from the chains of bondage, and attained to the world of liberty. They were freed from the vices of the human world, and were blessed by the virtues of the Kingdom. This is the meaning of the words of Christ, “I gave My blood for the life of the world”
6 —that is to say, I have chosen all these troubles, these sufferings, calamities, and even the greatest martyrdom, to attain this object, the remission of sins”