The quoted excerpt from Isaiah 53 above clearly occupies substitutionary language. This passage foretold Christ bearing the sins of mankind upon Himself. He was offered up in the same manner as an unblemished lamb for our sins (1 Peter 1:18-19). His innocent blood was shed for both the just and the unjust (1 Peter 3:18). We are healed spiritually by His wounds (1 Peter 2:24). The previously referenced passages from the New Testament are based on the vicarious nature of the atonement sacrifices performed under the Mosaic Law:
"When he finishes atoning for the holy place and the tent of meeting and the altar, he shall offer the live goat. Then Aaron shall lay both of his hands on the head of the live goat, and confess over it all the iniquities of the sons of Israel and all their transgressions in regard to all their sins; and he shall lay them on the head of the goat and send it away into the wilderness by the hand of a man who stands in readiness. The goat shall bear on itself all their iniquities to a solitary land; and he shall release the goat in the wilderness." (Leviticus 16:20-22)
Animals paid the price for the sins of people with their very own lives. They were an innocent substitute. Though animal sacrifices served as a temporary covering for sin, the Levitical sacrificial system pointed to the perfect once for all sacrifice of Jesus Christ.
It might also be interesting to point out that Jewish commentators in centuries past recognized Isaiah 53 as being a Messianic prophecy (along with the substitutionary language contained therein). Following are a few quotes from an article by Jews for Jesus titled Jewish Messianic Interpretations of Isaiah 53:
"Another statement from Yefeth ben Ali (10th c.):
By the words “surely he hath carried our sicknesses,” they mean that the pains and sickness which he fell into were merited by them, but that he bore them instead. . . . And here I think it necessary to pause for a few moments, in order to explain why God caused these sicknesses to attach themselves to the Messiah for the sake of Israel. . . . The nation deserved from God greater punishment than that which actually came upon them, but not being strong enough to bear it. . . God appoints his servant to carry their sins, and by doing so lighten their punishment in order that Israel might not be completely exterminated." (Driver and Neubauer, pp. 23 ff.; Soloff pp. 108-109)
"Herz Homberg (18th-19th c.):
The fact is, that it refers to the King Messiah, who will come in the latter days, when it will be the Lord’s good pleasure to redeem Israel from among the different nations of the earth…..Whatever he underwent was in consequence of their own transgression, the Lord having chosen him to be a trespass-offering, like the scape-goat which bore all the iniquities of the house of Israel." (Driver and Neubauer, p. 400-401)