Christ's Birth & His Identifying with the Sinner

Why are we given such details of Christ's first coming: his lowly birth, the fact that there was no room at the inn, that he was lying in a trough used to feed animals? These are given to us to show us that, from the beginning of his life on earth, Christ would identify himself with the sinner. His glory would be veiled in flesh, that He might become like us, yet without sin. He was wrapped, not in his glory, as was his due right, but in swaddling clothes. Is there another way that God could have bridged the gap between man and Himself? Yes, but it would have meant only JUDGMENT, well-deserved, for us! Thus the Lord Jesus Christ had to be born, identify with the sinner by his life, and die in the sinner's place.