Just because a person is willing to sacrifice their life to save others, doesn't mean that person doesn't fear death.
We all fear death. Every sentient being that ever lived, and ever will live, fears death. Anyone who says otherwise is a liar. But there are those who, even though they are afraid to die
, can put aside that fear in order to give their life to save others. Kirk is one of those people. So is Scotty. So are most of the regular characters we've seen.
As for Finney: he wants to make Kirk suffer, not die. The way to do that is to kill someone, or some thing, that is close to Kirk, so that Kirk would be forced to watch as that death occurs. That's worse than death, in a way.