It's complicated but from earliest days of the Jewish people they granted their God power to forgive much like people can forgive others. It is repeatedly stated so. Forgiveness was granted when repentance and change took place. It gets complicated as the cult services at the temple involved gifts of thanks,purification rites and covenant investment (eg. each person had to give half shekel to the priests as a cover price each census). In some instances, these rites are described in a way that many perhaps thought of the animal sacrifices as being a free pass, a mechanism resulting in forgiveness. Reformers like Jeremiah saw the system of animal sacrifices as being in a large way to blame for the 'lack of repentance' as he saw it. He goes so far as insist the whole idea of animal sacrifices was a scheme of lying priests/scribes
Go ahead, add your burnt offerings to your other sacrifices and eat the meat yourselves! 22 For when I brought your ancestors out of Egypt and spoke to them, I did not give them commands about burnt offerings and sacrifices, 23 but I gave them this command: Obey me, and I will be your God and you will be my people....actually the lying pen of the scribes.
Anyhow, his efforts were not successful for as soon as they could they returned to animal sacrifices.
It was a Christain innovation to connect a distorted view of animal sacrifices with the idea of a suffering/dying Messiah. It is true that the idea of a suffering messiah was established in Hellenized Jewish circles distanced from the Temple, before the assumed time of Christian origins. But the full development of the doctrine of God deliberately having his son killed for sins was a new idea best received by those Hellenized circles who laid much of the groundwork.