Brondos on atonement
"Generally, when it is said that someone has died or given up one's life "for" others, the meaning is that the person died as a consequence of the activity that that person was carrying out on behalf of others or in order to obtain some benefit for others. In this case, the idea would be that Jesus died as a consequence of his activity on behalf of others, in particular those who would come to be incorporated into the church after his death ("us"), or that his death would benefit them in some way. Both of these ideas can be brought together in considering Jesus' death: he dedicated his life to the kingdom of God and to laying the foundation for a new covenant to come about, and he refused to put an end to that activity when threatened with death, thus suffering the consequences of his activity for others; and by giving up his life in faithfulness to that mission, he obtained what he had sought for others when God raised him from the dead. Thanks to what he did in life and in death, as well as to God's response in raising him from the death, there is now a new covenant in which people may live and in which they may find assurance of salvation and forgiveness of sins. [...] His activity on behalf of others included both his work on behalf of the kingdom and the new covenant in which many would come to share as a result of his work, as well as the implicit petition made when he gave up his life that what he had lived and died for might become a reality." - Brondos, Paul on the Cross pp 109-110
Brondos' view, in short, is that Jesus' aim in life and death was to found the church, in the sense of creating a new covenant community. He persevered to the point of death in the face of opposition, petitioning God through his faithfulness to aid his cause, and God responded by raising Jesus from the dead.
In Col 1:24 Paul writes "in my flesh I am completing what is lacking in Christ's afflictions for the sake of his body, that is, the church." Brondos comments on this:
Jesus suffered "for" others in the same way Paul is said here to have been suffering "for" others: both were afflicted as a result of their efforts to establish and extend the community of believers now known as the church. Christ's sufferings had been "lacking" or insufficient in that the task of establishing the church throughout the world had not been completed when he died. (pg 111)