I believe the sense of dying for our sins (or above/around), is this:
I recall that some Fathers said that God would become man even if Adam had not sinned. Although Adam was in a state of purity, he was not yet deified, because human nature was not yet united with divine nature in the Son. The *main* purpose of the Incarnation is deification, the union of these two natures.
Now, Adam's sin causes death. It's beside the point here if it is a blessing or a curse. We simply know that it wasn't part of the game and now it is. So, God could have said: "I will not become one of you, now that you have brought suffering and death into human nature. I'm the God Almighty, I don't have to suffer or die!"
But He did not do that. He proceeded with the plan. He became one of us even though that would mean that the Purest, All-Holy, Almighty Creator of Everything beyond death and suffering in a complete absolute sense, would have to suffer and die.
So, because He loved us so much, He did not shun our sickly nature and wore it just like He would have if it still were the glorious nature without the Fall. He united Himself to us although we had become most impure, most undeserving. The best analogy is that of a loving groom whose bride abandon him for a life of prostitution with many men. They had the wedding date already set, but she simply forgot about it. On the set date though, the bridegromm comes regardless of everything the bride has done and, without requesting anything from her, he marries her just as nothing had happened declaring everlasting love. The bride, here, is human nature. And it is with this analogy that we understand that the salvation of our nature did start with that "Yes" of that young virgin bride so many years ago, the first of the Children of Adam and Eve to fully not behave as the pervert bride, the first to live as close as possible to that nature before the fall even though in an after-fall world.