I recommend jellyfin (https://jellyfin.org/). It will require some tech skills to setup, but after that accessing it is as simple as going to the website address it is hosted at.
The best thing about Jellyfin is that it is free software: open source, permissive license, and no paid features (which alternatives like Plex have).
You can host Jellyfin on your own VPS, I host mine on a Hetzner VPS for just over 10 eur / month but that is very overkill, you can host one for far less. The only thing to keep in mind is that transcoding (streaming in a different resolution to the stored media) is quite computationally expensive so if lot's of people are transcoding at once you will need a good CPU.
Then any family member can just visit the IP address of the server in their web browser (or you can buy it a domain name).
Jellyfin has a native iOS app they can use from their phones.
For viewing it on TVs, there is an app for AppleTV which has Jellyfin support (cant remember the name right now but I can find it if you need it).
The best thing about Jellyfin is that it is free software: open source, permissive license, and no paid features (which alternatives like Plex have).
You can host Jellyfin on your own VPS, I host mine on a Hetzner VPS for just over 10 eur / month but that is very overkill, you can host one for far less. The only thing to keep in mind is that transcoding (streaming in a different resolution to the stored media) is quite computationally expensive so if lot's of people are transcoding at once you will need a good CPU.
Then any family member can just visit the IP address of the server in their web browser (or you can buy it a domain name). Jellyfin has a native iOS app they can use from their phones. For viewing it on TVs, there is an app for AppleTV which has Jellyfin support (cant remember the name right now but I can find it if you need it).