I think the reason immortality is scary to a lot of people boils down to an ingrained belief that life has 'stages'.
You spend your formative years learning how to be a person, until you're in your mod twenties where you know, and now you start with your Life.
For the next thirty years or so, you contribute to society, you learn more, you play, you work, you try not to work too much, but hey, it happens. You love, you get a partner, you get a few, until a certain point where you decide you're done with the 'work' part of life. After that, you focus on joy, play, love, whatever you can with retirement (obviously, this is not the default, just the ingrained ideal of western society).
After you're sufficiently done and your body is broken, you settle down with someone, or someone's, and wait to die. Throughout all of that, people you love die, but it comes with an understanding that that's just a part of life, and soon once your stages are done, you'll die too.
Immortality is… daunting in that. Most immortality horror, the truly good ones, don't stick with 'everyone I love dies', and instead focus on the fact that life without stages is truly, truly boring. Do you work forever, or play forever? Both will get old, right? And being immortal, you're responsible for yourself. There's no elder, no teacher, you can't look to someone older than you to see how they're doing in their late 1400s.
It's forever, it's structure less, and it's completely, utterly independent, and without a good structure, that can be scary.