I will structure my answer in two parts. First I will focus on why the he-goat for the LORD had no correspondence in Jesus' sacrifice. Then I will focus on why the scapegoat was a figure of Jesus.
Part 1. Why the he-goat for the LORD had no correspondence in Jesus
The rites carried out by the High Priest in the feast of Atonement involved three animals: a bull and two he-goats, one for the LORD and one for azazel (meaning "entire removal"). The High Priest would ...
slaughter the bull to make atonement for himself and for his household (Lev 16:11). This had no correspondence in the case of Jesus, since He was sinless.
slaughter the he-goat for the LORD as a sin offering which is for the people, and
take some of the blood of the bull and then of the blood of the he-goat and sprinkle it "on the mercy seat and in front of the mercy seat" (Lev 16:14:15) to "make atonement for the Holy Place" (Lev 16:16), then on the tent of meeting (Lev 16:16), and then on "the altar that is before the LORD" to "make atonement for it" (Lev 16:18), specifically to "cleanse it and consecrate it from the uncleannesses of the sons of Israel" (Lev 16:19).
Thus, the he-goat for the LORD was sacrificed not to cleanse the sons of Israel, but to cleanse the Holy Place and the altar from the uncleannesses of the sons of Israel. But in the real Atonement carried out by Jesus the Holy Place and the altar were his Most Holy Body, which needed no cleansing. Therefore, the he-goat for the LORD had no correspondence in Jesus' sacrifice.
Part 2. Why the scapegoat was figure of Jesus
This explanation is based on the passage in John's Gospel when the people that had come to apprehend Jesus fall to the ground when He said "I Am" for the first time (Jn 18:5-6). To apprehend the theological meaning of this passage we need to take into account three data items:
First, the isolated statement "I Am" (Ego Eimi), with which Jesus identifies Himself twice in the passage, appears in 4 previous verses in John's Gospel:
«for unless you believe that I Am, you will die in your sins.» (Jn 8:24)
«When you have lifted up the Son of Man, then you will know that I Am, and that I do nothing on my own;» (Jn 8:28)
«Truly, truly, I say to you, before Abraham was, I Am.» (Jn 8:58)
«I am telling you now before it happens, so that when it comes to pass, you may believe that I Am.» (Jn 13:19)
In all these verses it is clear that Jesus applies to Himself the proper Name of God in the first person revealed in Ex 3:14: Ehyeh, "I Am". This is particularly evident in the first, second and fourth verses, in which Jesus notes the importance of {beleiving/knowing} that "He Is", echoing Isaiah 43:10 y 48:12.
Second. The Greek term "fell" (epesan) is used 5 times by the Apostle John in Revelation in the sense of "fall on their face" to worship: 5:8, 5:14, 7:11, 11:16 and 19:4.
Third. By the time of Jesus, the proper Name of God in the third person revealed in Ex 3:15: YHWH, "He causes to be" if vocalized YaHWeH, was uttered by only one person, the High Priest, on only one day of the year, the feast of Atonement (Yom Kippur), 6 times when making a sacrifice for his own sins, one time when drawing the lot for the he-goats, and 3 times when loading the iniquities and transgressions of the sons of Israel on the he-goat to be sent to the desert (Lev 16:20-22). The prayer used by the High Priest for the latter function, and the people's response, are in the Mishna, tractate Yoma, chapter 6:
He then came to the he-goat which was to be sent away to
Azazeil and forcefully leans his hands on it and confesses. And so he
would say: Please O YHWH, they have done wrong they have transgressed
they have sinned before You - Your nation the House of Israel, Please,
O YHWH, forgive them for their doing wrong, for their transgressions
and for their sins, as is written in the Torah of Moshe Your servant:
“For on this day He will effect atonement for you to purify you before
YHWH” (Leviticus 16:30). And when the priests and the people who were
standing in the courtyard heard the fully pronunced Name come from the
mouth of the High Priest they would kneel, prostate themselves, fall
on their faces, and call out: Blessed be the Name of His glorious
kingdom for ever and ever. He gave it over to the one who was to lead
it [to Azazeil].
Mishnah Yoma, chapter 6
Mishnah Yoma 6
From these data, the meaning of the fall to the ground of the party that had come to apprehend Jesus when He said "I Am" for the first time is crystal clear: Jesus is the High Priest who is carrying out the true Atonement prefigured by the rite in the Mosaic Law, and that at the time of loading the iniquities and transgressions of men on the victim that will carry them, pronunces the proper Name of God, with the difference, with respect to an ordinary High Priest, that:
since Jesus Himself is the victim, He bears and carries our iniquities and transgressions Himself,
since Jesus Himself is God, He pronounces the proper Name of God in the first person.
Finally, the third time when Jesus pronounces the proper Name of God in the first person as true High Priest of the true Atonement is not recorded in John's Gospel but in Mark's, in the reply to the High Priest
Again the high priest was questioning Him, and says to Him, "Are You
the Christ, the Son of the Blessed One?" And Jesus said, "I Am. And
you will see the Son of Man sitting at the right hand of Power and
coming with the clouds of heaven." (Mk 14:61-62)
To validate the interpretation of this "I Am" as the third uttering of the divine Name in the first person by Jesus as High Priest of the true Atonement, we must note that, in the Jewish rite, immediately after the High Priest finished his prayer uttering the divine Name by a third time, the goat was taken to the desert. Similarly, immediately after pronouncing the third "I Am" in Mk 14:62, Jesus started to be spit, striken, mocked and slapped by the Jews (Mk 14:65).
Acknowledgment: I learned of this theological meaning of the passage from a site on the revealed Name of God by a Jewish scholar:
THE NAME OF GOD AS REVEALED IN EXODUS 3:14
Give it a look over. very interesting
– Kris Nov 04 '15 at 23:06