I feel that I more trust some corpo (Google, etc) that one particular person.
I don't imagine setup where you can effictevely guarantee them full privacy.
Some people in my family use it as their main address, others don't, it's entirely their call.
But yes, ultimately I control the domain and could be nefarious if I wanted to. But there's a certain baseline level of trust as a family, I'm reasonably certain my wife won't poison the milk in the fridge and she's reasonably certain I'm not going to read her emails.
I believe this is a very common setup: the "computer wizard" kid of the family manages the computers for the whole family. Not just emails, they have access to the whole computer (and have to fix when it breaks).
But it hardly needs to be difficult. If you're running dovecot and postfix on a server somewhere then yes, family is screwed. But it's simple to use either some mail forwarding service that you pay for with a credit card, or something like fastmail (etc). Leave 2 pages of instructions for how to log into and renew the domain (print the QR code used for the 2fa enrollment!) and how to log in and pay for whatever the underlying services are. Place in a binder and label "Family.Name Email Management" and put it with your other important documents.
But more seriously, my domain and VPS is on auto-pay, so it doesn't just shut off the day I die. My survivors will have plenty of time to back up their emails and do whatever they want with them afterward.
Plus, the password for my computers and keychain is in a safe-deposit box if they feel like handing it over to a trusted tech-savvy friend of the family to shut down properly.
It made sense to us because our starting point was an email service letting people share lastname.sometld, but we never got close to as many registrants on .name as we had users on the webmail service (we had a couple of million accounts on that when it was sold to one of Marc Cubans companies for a relative pittance in the aftmath of the dot com bubble bursting)