You know a little about Buddhist monks. Some of them come to cherish life so much that to step on an insect or harm a blade of grass becomes a violation of their creed, so the place themsleves under an extreme form of protective custody, a sanctuary of sorts. Well, for an Immortal who comes to abhor the bloodshed, there's a similar solution - a way to be removed from the game forever. Though the price is unimaginably high, but you are, for all practical purposes, protected from the violence within you.
You're missing the point, Kate. The difference between Connor and I is that as long as you're still alive, there's a chance that one day I might be forgiven. It may take years. Centuries even. But at least I can carry that hope inside me. That's one blessing of immortality; there's always tomorrow. Even for us.