I pretty much rationalize away any fear that I have. I've seen people live their lives in fear and it's not something I ever want to do. Everything carries a risk, everything is taking a chance. If I really lived in fear of death I wouldn't drive or ride in a car because that is hella unsafe.
Like Marm said, most people are afraid of how they might die or of pain but that's not even being afraid of death. I also agree with Qlippoth in that it seems so far away, is my kindergartener afraid of graduating from high school? When it gets to the point of dealing with death you've already lived your life. I imagine when someone knows they will die there may be real rational fear, otherwise it's too "far away" to worry about.
I have very few fears just because of not wanting to live in fear. As a teen I was afraid of dying and I was afraid of rape. I didn't want my first sexual experience to be rape and I didn't want to die before I experienced marriage or having children. I'm not as afraid of death anymore. It's a natural thing that happens, we are small parts of a large whole of humanity.
I don't want to die because I want my kids to know me and I want to know my grandkids but otherwise no, I'm not afraid of death at all.
There is a difference with being "afraid" of something and having a fear of it. I can get freaked out by nasty bugs but bugs don't "scare" me.
I'm guessing more people are afraid of pain than of death.
Erm: This seems like a chronic fear - from the OP "deeply unsettled by the thought."