Let me try an answer on you that makes some sense to me...
Here in Matthew 19 Jesus is recorded as asking the impossible of a really nice guy...
16And, behold, one came and said unto him, Good Master,
what good thing shall I do, that I may have eternal life? 17And he said unto him, Why callest thou me good?
there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.
18He saith unto him, Which? Jesus said, Thou shalt do no murder, Thou shalt not commit adultery, Thou shalt not steal, Thou shalt not bear false witness,
19Honour thy father and
thy mother: and, Thou shalt love thy neighbour as thyself.
20The young man saith unto him, All these things have I kept from my youth up: what lack I yet?
21Jesus said unto him, If thou wilt be perfect, go
and sell that thou hast, and give to the poor, and thou shalt have treasure in heaven: and come
and follow me.
22But when the young man heard that saying, he went away sorrowful: for he had great possessions.
23Then said Jesus unto his disciples, Verily I say unto you, That a rich man shall hardly enter into the kingdom of heaven.
24And again I say unto you, It is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God.
You could argue that the message here was that the decent young guy in the encounter was wanting to do his best, and Jesus went straight to the point and showed him that he needed to be perfected by giving away all his goods in order to have eternal life. You might then feel inclined to pick up on Jesus saying there is none good but God. Again, us Jesus saying he is not God or is he trying to provoke this man to think?
The standard set was so high that the disciples thought to themselves 'If this great guy can't do enough to make it, what hope is there for us regular Joes' (my paraphrase)
25When his disciples heard
it, they were exceedingly amazed, saying, Who then can be saved?
26But Jesus beheld
them, and said unto them, With men this is impossible; but with God all things are possible.
In the opener to this passage our man asks the million dollar question - what
good thing shall
I do, that I may have eternal life...
This guy was asking what could be done to make his works measure up. The answer asked too much of him.
Getting to your question, how can someone pay for the sins of another? If I am in debt and I ask someone else with no cash to pay off my credit cards, the bank won't be very interested. The only way to sort that situation out is for someone with a decent bank balance to pay off my creditors for me.
Perhaps the OT restriction on paying for anothers misdeeds falls under this pattern. A tradition that provokes a question. Similarly, as the Jews sacrificed a lamb without spot or blemish at passover the question must have resonated with them over the years 'why'. For some, when Jesus gave his life as meekly as a lamb at Passover, they would have made that connection.
You could remain in the same position as the young man in the account, saying "what good thing shall I do, that I may have eternal life
". I am not quite so confident of my righteousness, I guess I identify more with another guy from Luke 18:3
"But the tax collector stood at a distance. He would not even look up to heaven, but beat his breast and said, 'God, have mercy on me, a sinner.'