“A person who imitates another is in a very real sense acting in the person of another, rather than in the presence of another.”
In the previous verse, 15, Paul says the Corinthians had many tutors but few fathers “en”, in Christ.
There were actually two contrasts, tutors and fathers, the many and the few. Nothing about the Corinthians “merely listening” but rather Paul's fatherly treatment of the Corinthians versus how a tutor might.
While Paul COULD say he was acting in place of, or in stead of Christ, that is not the meaning of the word “en”.
Nor is it what Paul said at 2 Cor. 2:10, his ability or authority to forgive sins was not the context but rather his willingness to follow the decision of the Corinthians.
Do not confuse them with context
To act in the person of Christ means to have the authority of Christ to decide the matter on hand, as if Christ Himself had decided it. This is clear from John 20, where Christ said “as my Father sent me, I send you. What sins you forgive, will be forgiven in heaven, and what sins you retain will stay retained”.
This might explain the distinction between teaching and fathering. While both are important, the “fathering” in 1 Cor. 4:15 is, I think, a reference to the spiritual authority Paul has as an apostle, that is, as someone sent by Christ as Christ was sent by His Father. But I would concede that it is not plainly visible in the text, it is just how I understand it.
Now, in 2 Cor. 2:10 the situation is complicated because the local church already pardoned the penintents, and now St. Paul attaches the authority of Christ to it. This is also the understanding of St. John Chrysostom in the homily cited before, and I think it is plain in the text.