For a privacy friendly OS, surprised nobody has mentioned Freedombox
It’s designed explicitly for your use case, along with an easy path to other self hosted services. When you’re ready for more than it offers through the web interface, it’s a full Debian install under the hood - so you can install whatever you need to. Privacy friendly and super stable, with smooth upgrades to new releases and security updates for old versions several years after the new one is available.
As far as hardware, your old computer is probably more powerful than a Pi and can support more drives, but the Pi will be more power efficient. As others have mentioned, if you care about your data long term then backups are a must, so a separate NAS or a Pi with a large drive for backup storage is a good idea as well, whatever OS you choose.
Sorry, but I have bad news for you. Privacy in major car brands no longer exists.
You don’t say where your family member lives, but you might look into smaller regional brands that focus on cheap cars for less overdeveloped areas of the world. Be aware the tradeoff is probably in safety features.