The family might be interested in revenge. Society has an interest in all three other possible motivations, though probably not in revenge.

This is the crux of the whole issue. The US is split between the two notions of justice: retribution and rehabilitation. The split seems to largely fall along political lines, which means the argument isn't likely to be settled any time soon.

I tend to fall in the rehabilitation camp but I can see some of the logic behind the other side. At the very least, I recognize that it's better for the state to apply measured retribution than for families to pronounce blood feuds upon one another. Beyond that, I'm at a loss for what to do about cold, psychopathic individuals who seem utterly incapable of reform.

Someone can agree with both of your views depending on the crime. Physically harming another person, particularly irreparably is different than stealing something for example.

If a patient goes to a doctor and their wound heals, is that "acting healthy"?

You can make a case that we can't know if someone is rehabilitated, but if you think rehabilitation can be a thing, that necessarily includes being able to distinguish acting from genuine change.