Currently, we're just using Google Keep, since it at least supports syncing, sharing, and checklists, but I'm hoping to find something more privacy friendly, and/or more feature rich. In particular, what I'm looking for, in order of priority, are as follows:
Doesn't need to be self-hosted
Lists that can be shared and synced across users and devices
Basic Checklist functionality
Reminders, ideally with options to repeat until complete or similar
Automation, such as automatically recurring events, esspecially recurring after date of completion
Private storage of user data, or at least better than Google
I recommend you to take a look into Single Board Computers (SBCs) like Raspberry Pi, Orange Pi, Le Potato, Rock64, etc. They are very capable of hosting most things that ordinary people use, at the scale of a family. There are many guides online on how to make a personal VPN of it, as well as host things directly on it like your own private file share and sync.