Good question. I think the payment idea is not best understanding of the Bible. Jesus gave right to forgive sins to his disciples and he also forgave sins before he died. That is why I think death is not necessary for to forgive sins.
The scribes and the Pharisees began to reason, saying, "Who is this that speaks blasphemies? Who can forgive sins, but God alone?" But Jesus, perceiving their thoughts, answered them, "Why are you reasoning so in your hearts? Which is easier to say, 'Your sins are forgiven you;' or to say, 'Arise and walk?' But that you may know that the Son of Man has authority on earth to forgive sins" (he said to the paralyzed man), "I tell you, arise, and take up your cot, and go to your house." Immediately he rose up before them, and took up that which he was laying on, and departed to his house, glorifying God.
Luke 5:21-25
Whoever's sins you forgive, they are forgiven them. Whoever's sins you retain, they have been retained."
John 20:23
So, if person makes mistakes, disciples of Jesus have the right to forgive them. And there is no requirement of sacrifice.
But, it can be said Jesus died because of our sins, when it means, he died, because he did the job and declared words that God had commanded him to say, words that can give the forgiveness.