Did the earliest Christians believe in substitutionary atonement and imputation?

Did Jesus die in our place, bearing the punishment for our sins? In Did the Earliest Christians Really Believe in Substitutionary Atonement (and Even Imputation)?, Dr. Michael J. Kruger writes:

The average internet-level narrative goes something like this: the earliest Christians had no clear understanding for why Jesus died on the cross and what it accomplished. The idea of a substitutionary atonement is a late invention designed to retroactively explain the (otherwise embarrassing) death of Jesus. In fact, it was not until Anselm’s Cur Deus Homo (Why the God-Man?) in the middle ages that someone came up with the idea that Jesus died in place of sinners.

Of course, such a narrative can be readily refuted just examining the writings of the New Testament itself–particular[ly] the letters of Paul. However, it is also worth noting that this view was held by some of the earliest Christian writers; in this case, by the author of the Epistle to Diognetus in the early second century. Here are some excerpts from the author that affirm key aspects of substitutionary atonement: