Right off the bat I gather there are issues with 1 Cor. 15:3-8 and Php. 2:5-11, but all things considered they seem like good enough starting points for me.
For what I received I passed on to you as of first importance: that Christ died for our sins according to the Scriptures, that he was buried, that he was raised on the third day according to the Scriptures, and that he appeared to Cephas and then to the Twelve. After that, he appeared to more than five hundred brothers at once, most of whom are still living, though some have fallen asleep. Then he appeared to James, then to all the apostles. And last of all he appeared to me also ...
Let this mind be in you which was also in Christ Jesus, who, existing in the form of God, did not consider equality with God something to be grasped, but emptied Himself, taking the form of a servant, being made in human likeness. And being found in appearance as a man, He humbled Himself and became obedient to death—even death on a cross. Therefore God exalted Him to the highest place and gave Him the name above all names, that at the name of Jesus every knee should bow, in heaven and on earth and under the earth, and every tongue confess that Jesus Christ is Lord, to the glory of God the Father.
In the big picture I'm inclined to see this kind of talk as Fourth Philosophic nonsense, but the idea seems to be that Jesus was the Messiah (Christ) and as such he was equal with God and the embodiment/fulfillment of Isaiah's Suffering Servant because of his suffering and death, as per Is. 53:4-6:
Surely He took on our infirmities and carried our sorrows; yet we considered Him stricken by God, struck down and afflicted. But He was pierced for our transgressions, He was crushed for our iniquities; the punishment that brought us peace was upon Him, and by His stripes we are healed. We all like sheep have gone astray, each one has turned to his own way; and the Lord has laid upon Him the iniquity of us all.
This passage is referred to in 1 Peter (2:21-25), which I consider to be genuine (and thus pre-70 CE).
For to this you were called, because Christ also suffered for you, leaving you an example, that you should follow in His footsteps: “He committed no sin, and no deceit was found in His mouth.” When they heaped abuse on Him, He did not retaliate; when He suffered, He made no threats, but entrusted Himself to Him who judges justly. He Himself bore our sins in His body on the tree, so that we might die to sin and live to righteousness. “By His stripes you are healed.” For “you were like sheep going astray,” but now you have returned to the Shepherd and Overseer of your souls.
So in my view the idea that Jesus' death atoned for sins is early and genuine and was derived largely from Isaiah's Suffering Servant, but how does this atonement work in Christianity? My understanding is that the Suffering Servant figure does not negate the necessity of making sin offerings in accordance with the Torah, and Jesus (in what I consider to be the earliest gospels, Mark, Matthew and Luke) and early Christians (in Acts) do not appear to have thought that sacrifice was obsolete.
So what does it mean in this context that Jesus "died for our sins"? For me it is the same as asking, What does it mean that the Suffering Servant "died for our sins"? Neither figure (the Suffering Servant and Jesus) appears to me to negate the necessity of making sacrifices in accordance with the Torah. So in what sense (or to what degree) then do they serve as sin offerings?
Perhaps they were alternative sin offerings and not replacements. I gather that the context of the Suffering Servant is as a symbol of Israel in exile, when sacrifices could not be made in the Jerusalem Temple. And in the first century CE I gather that the priestly service was viewed by some as being corrupt or ineffective (or a "den of robbers" in Jesus' case). Yet neither context appears to negate the necessity of making sacrificial offerings in accordance with the Torah. In the first, sacrifices will resume when the Exile is over, and in the second, some Christians and Essenes offered sacrifices despite whatever reservations they had about the priesthood or beliefs they had about alternative sin offerings.
In these contexts, I gather the suffering and death of the Suffering Servant and of Jesus were viewed as alternative sacrifices in an imperfect world (exile, corrupt Temple service), in the same way that prayer and charity serve as alternative means of atonement in Judaism in the absence of a Temple, without negating the necessity of making sacrificial offerings in accordance with the Torah.
When sacrifices were offered in ancient times, they were offered as a fulfillment of Biblical commandments. Since there is no longer a Temple, modern religious Jews instead pray or give tzedakah instead to atone for their sins as the korbon would have accomplished. According to Orthodox Judaism, the coming of the messiah will not remove the requirement to keep the 613 commandments, and when the Temple is rebuilt, sacrifices will be offered again.
https://en.wikipedia.org/wiki/Korban
So it appears that if you can't (or don't want to) go to the Temple (because it doesn't exist, or you live too far away from it, or you think its service is corrupt), then you can pray and/or give charity and/or believe that Jesus was the Messiah to atone for sin, and at the same time you can also believe in the necessity of making sacrifices in accordance with the Torah.
In other words, it looks to me like prayer, charity, the Suffering Servant and Jesus are alternative (or additional) means of atonement rather than replacements for the Temple service. This is why according to what I view as being the earliest gospels (Mark, Matthew and Luke) and the earliest Christian history (Acts), Jesus is pro-sacrifice and Christians continue to offer sacrifices after Jesus' death.
But setting aside how prayer, giving charity, and making sin offerings in accordance with the Torah "works" (which equally baffles me), how does belief in Jesus as the Messiah "work"? If you don't believe Jesus' death was a sin offering then it doesn't work (and Jesus judges you accordingly when he returns as a world conquering spiritual being at the End Time), and if you do believe it then your belief somehow activates the efficacy of the sin offering (and Jesus consequently likes you when he returns as a world conquering spiritual being)?
I suppose I could ask how does Jesus know who believes in him and who doesn't, but since I gather he was thought to be equal to God, then he would know via his God-like ability. And the same goes for his death being an alternative sin offering; it "works" via "the power of God," the same way that making sin offerings in accordance with the Torah "works." In other words, it's nonsensical.