Pixelfed is already what you are looking for. You could contribute yourself, fork it, or just use it in your server.

What I don't see often, and I think it was a neat idea is a copy of Path. A social network for a very very small set of people to share their things. I think path was like 2 members.

I imagine that just a whatsapp/telegram conversation chat is more than enough for this case tho.