In the ancient world, Jews, Egyptian, Latin, Greek, Pagan, the standard way of worship was the sacrifice (as today the hymns and the boring sermons..).
It is reasonable that Jesus introduced many new teachings, but it is unreasonable that he requested a complete change of the religious attitude of the time.
Actually Jesus never excluded sacrifices: He asked to move from the bloody sacrifices of animals that happened in the court of the Temple (in open air) to the un-bloody sacrifices that the priests made in the nave of the Temple: the sacrifice of frankincense (read as the offering of our prayers) and the weekly sacrifice of the breads (the model of the sacrifice of the Holy Eucharist)
No, Jesus never dismissed the cult of the Temple as being about the money. He never attacked the cult in the Temple. "He said to them, "'My house will be a house of prayer'; but you have made it 'a den of robbers.'""
Jesus is not against the cult of the Temple, but he is against who, because of money, do not respect the holiness of the place.
You can find the sacrificial meaning of Jesus death also in others than Saint Paul: consider for example in the Revelation the Lamb of 5:6-9 which is surely an image of Christ, being he adored and worshiped with the Father on the throne (no other figures can be adored and worshiped with the Father other than Jesus himself, being the Holy Spirit be represented by the seven eyes of the Lamb)
For sure the sacrificial meaning of Jesus' death is not only typical of Saint Paul only
Can Jesus forgive without sacrifice?
According to the classical Catholic theology (that here differs from the later Protestant theology): It was not necessary, then, for Christ to suffer from necessity of compulsion, either on God's part, who ruled that Christ should suffer, or on Christ's own part, who suffered voluntarily. Yet it was necessary from necessity of the end proposed (St Thomas ST 3,46,1)
And more: But if He had willed to free man from sin without any satisfaction, He would not have acted against justice. For a judge, while preserving justice, cannot pardon fault without penalty, if he must visit fault committed against another--for instance, against another man, or against the State, or any Prince in higher authority. But God has no one higher than Himself, for He is the sovereign and common good of the whole universe. Consequently, if He forgive sin, which has the formality of fault in that it is committed against Himself, He wrongs no one: just as anyone else, overlooking a personal trespass, without satisfaction, acts mercifully and not unjustly. And so David exclaimed when he sought mercy: "To Thee only have I sinned" (Psalm 50:6), as if to say: "Thou canst pardon me without injustice." (St Thomas ST 3,46,2)
Actually the translation you quoted already gives a pre-defined interpretation (using the preposition "that"), an interpretation that is in line with Luther sacrificial though, but which is not the only one possible.
The Greek for "that takes away" is "ho airon". The point to to who the "ho" ("that or who") refers to. To the Lamb or to Jesus himself ?
In Greek it is possible that "ho" refers or to the Lamb but also to Jesus.
The protestant translations, using "that" in place of "who" (or sometime using "which") gives the idea that it is the Lamb, not Jesus, who takes away the sins, thus requiring the sacrifice.
This use in the Protestant Bible cames from Luther translation in German where he used a neuter-gender pronoun that can be applied only to the Lamb, not to Jesus. Luther, further, falsified his translation translating "airon" ("take away") with ("carry"), in order to force the reading of this verse as a reference to the famous Is 53:4-7 "Surely he hath borne our griefs, and carried our sorrows:.... He was oppressed, yet he humbled himself and opened not his mouth; as a "lamb"(seh) that is led to the slaughter, and as a "sheep" (rachel) that before her shearers is dumb"
Usually Christian translation of Is 53:7 translate the Hebrew "seh" with "lamb" in order to underline the parallelism with Jesus's end, but the Hebrew term is a general term for "sheep" (male or female, young or old), while the next "rachel" means ("sheep which is mother").
And anyway the "lamb that is led to the slaughter" of Is 53:7 does not by itself include the idea of sacrifice, because it was normal for the Jews to slaughter sheep simply to eat them.
I end this boring technical post suggesting that not only "who take away the sins" is referred to Jesus and not to the "Lamb", but also suggesting that the "Lamb" of John 1:29 does not refer to Is 53:7, but it refers to Is 40:11 "He shall feed his flock like a shepherd: he shall gather the lambs with his arm, and carry them in his bosom, and shall gently lead those that are with young.": in John 1:29 Jesus is the little lamb in the bosom of the Father ! (the reference to Is 40 is common in the Gospel about Jesus baptism: compare Lc 1:15 with Is 40:3)
Nevertheless there are other passages in the NT where Jesus is clearly compared to the Lamb of Is 53:7, for example Acts 8:30-32)
