How do you pay off/get rid of someone else’s sin(s)?
By God accepting whatever it is that He accepts for it.
I don’t see why you’re having so much trouble with this. Teshuva, by it’s very nature, is a personal thing. Having someone do teshuva for you makes about as much sense as having some sleep for you. It can only be done by the person himself. The whole point is to transform and change the person for the better.
Kapparah doesn’t change you. It just removes a part or all of someone’s sins — but it doesn’t change the person’s behavior or attitudes.
The Wolf