Even within Christianity, the Atonement is, in part, a mystery. I learned that from my father-in-law, who is a world-famous theologian. When I spoke to him, I thought that atonement as "at-one-ment" was a bad pun, but he actually liked the idea.
Atonement in Judaism was the result of offering a sacrifice. But the core is not sacrifice, the core is what happens to the sacrifice, how it is uplifted and made holy.
In relation to the life of Jesus, I think that the Resurrection is more important than the Crucifixion. The Resurrection is the promise to all of us that no matter what we suffer in life, and no matter bodily death, there is an Eternity of Peace awaiting us.
Atonement is the process that brings us to that peace. Or, perhaps, brings that Peace to Earth through us.
And Atonement is not a part of Western religions only. In Buddhism, we recognize our errors - greed, fear, confusion - and offer them up, and allow them to be uplifted and transformed into Love and Wisdom. Then we live bringing Love and Wisdom into the world.
That is, I believe, Atonement.