These are not minor details. These actions describe the primary role of the live goat. Their fulfillments should be easily found in God's Word - and indeed, they are. These two identifying actions are found in the Messianic prophecy in Isaiah 53:
Verse 6 gives a clear fulfillment of Leviticus 16:21. The Eternal laid our iniquities on the Messiah, just as the high priest laid his hands on the azazel and confessed over it all the iniquities of the people.
In general, the laying on of hands indicates a solemn identification, a testimony, or a setting apart. It frequently contains the idea of transference. In the sin offering, the laying on of hands symbolizes the identification of an innocent substitute to whom sin is transferred from the guilty party. The hands also identify who is to bear the sin.
Now, the live goat was a substitute. It was not guilty, nor was it being blamed for sin. Instead, the sins of the nation were symbolically transferred it, and it bore them away from God's presence in the Tabernacle. The purpose of a substitutionary sacrifice is to have an innocent representative standing in the place of a person or group, so the guilty party does not have to bear the sins. The animal stands in for the sinner(s). The azazel was the type in receiving the sins of the nation, and the Messiah was the antitype in receiving our sins.
In verse 4, the Messiah is prophesied to bear our griefs and sorrows, which are not the sins themselves, but which are the effects of sin. Verses 4-5 portray the trauma the Messiah would undergo. They foretell that the Messiah would do more than just die. If God only required death for His justice to be satisfied, He could have had the Romans cut Christ's throat, just like an animal's. One quick and deadly slice, and it would be over. Yet Isaiah foretells that the Messiah would undergo incredible suffering before death.
There is a potent lesson here, which is that sin incurs more than just the death penalty. Sin also causes physical and emotional pain. It causes grief and sorrow. It causes separation between people, and more critically, between mankind and God, beginning with mankind's expulsion from God's presence in the Garden of Eden (Genesis 3:22-24; see Isaiah 59:1-2).
There is much that could be said about all the rotten fruit that sin produces, but for our purposes, it is enough to recognize that when God laid our iniquities on the Messiah, that action caused more than death. It caused unparalleled agony and disfigurement.
Isaiah 52:14 says that Christ was marred more than any man, such that it was hard to tell He was even human. That is what sin does - it distorts and corrupts the image in which mankind was created. We were created in God's image, but sin destroys that likeness.