I’m not sure that this is a given; that we would have sinned even if Adam and Eve had not sinned. A consequence of original sin is concupiscence, the tendency toward sin. Our intellect was dimmed and our will weakened so that the flesh became stronger than the will. Paul points this out in Romans 7:14-19:
"We know that the law is spiritual; but I am carnal, sold into slavery to sin. What I do, I do not understand. For I do not what I want, but I do what I hate. Now if I do what I do not want, I conqur that the law is good. So now it is no longer I who do it, but sin that dwells in me. For I know that good does not dwell in me, that is, in my flesh. The willing is ready at hand, but doing the good is not. For I do not do the good I want, but I do the evil I do not want."
On the other hand, Adam and Eve were not living under the consequences of original sin when they chose to sin, so it is quite possible, and maybe even probable, that we would have fallen into sin anyway. It’s a very interesting question. When we reach heaven we do not become robots, but rather we remain beings with a free will, which would mean that we are still free to choose sin, yet we will not sin. So it is certainly possible to live with free will and not sin. Mary is a perfect example of this. She was born free from the consequences of original sin and remained sinless her entire life.
The primary purpose of Christ’s incarnation was to defeat sin and death by offering the perfect sacrifice to the Father and redeeming mankind. He came to conquer Satan. As others have pointed out, however, this was not his only purpose. We read that Adam and Eve walked with God in the cool of the day. If they walked with God, they walked with Christ, so I think it is quite possible that he would have accomplished his other purposes had we remained in the Garden.
This is a great question, but I think the best we can do at this point is to speculate.