According to a ‘forgiveness in Christ alone’ framework how did ‘sin offerings’ make atonement for sin under the Mosaic Law? How did the old ceremonial sacrifices made atonement? What was really going on externally and internally to the person who brought the ‘sin offering’ to God? What did it mean that their ‘sins were atoned for’? Did it make a difference if the sacrifice was made by faith or not?

Note: I am specifically looking for an answer under a ‘protestant framework’ that assumes all real forgiveness must be by ‘faith in Christ alone’.
How did the old ceremonial sacrifices made atonement?


They didn't. Old Testament sacrifice was only able to atone for someone who followed the Mosaic law perfectly... a feat which is beyond any of us, especially as most of us are not Jews, and even if we were the Altar and priests are gone. The whole point of Jesus' sacrifice is that the former sacrificial system was inadequate. Sin offerings succeeded only until "just the right time", when Christ made one sacrifice for all.