Paper is a system that would have worked 30 years ago, unchanged to today. I think there's a reasonable chance that the same system would be working, unchanged, 30 years from now.
How many computer-based systems from 30 years ago are working unchanged to today?
Also, it's a system that any member of my family at any time in the past half-century from ages 12 to 98 could have understood. I think there's a reasonable chance that the same technology (paper & boxes) will be understandable for the next half-century, as well.
If you want to be updating a computer-based system for something as important as this every 5 years, that's fine. Everyone needs a hobby. But I have no guarantee that my family will be interested in or comfortable with it when it is needed.
I don't think I'd go as far as saying I'd never rely on a computer-based solution, but I am weary of putting all this extremely important information in a potentially vulnerable place. As a lot of people here are saying, paper might just do the trick.
So I guess for your purposes this means focus heavily on security. Keep us updated on how it goes!
"I would never rely on a computer-based solution for this."
Could I know precisely why ? (your answer could help a lot).