How, Thomas asks (ST, III, 48, 1), can Christ earn salvation for other people? He answers by reference to the totus Christus: “Grace was bestowed upon Christ, not only as an individual, but inasmuch as He is the Head of the Church, so that it might overflow into His members; and therefore Christ’s works are referred to Himself and to His members in the same way as the works of any other man in a state of grace are referred to himself. But it is evident that whosoever suffers for justice’s sake, provided that he be in a state of grace, merits his salvation thereby . . . .Consequently Christ by His Passion merited salvation, not only for Himself, but likewise for all His members.”
How, he asks (III, 48, 2), can Jesus’ atone for sin when He’s never sinned, and therefore can’t confess or show contrition? His answer is about the body: “The head and members are as one mystic person; and therefore Christ’s satisfaction belongs to all the faithful as being His members.”
For Paul, the body of Christ is not “merely” an eccesiological fact but a soteriological reality. Without it, we can’t make much sense of the atonement.