There's nothing conflicting about it. It's not a public timeline, it's "public" only to people you've added, no one else, including the server that would host your content.
Basically old Facebook (sharing just to your friends), without the spying, is what I'm asking for.
You would manage who can access what, by allowing/not allowing people to follow you
It's not a group abstraction, at least for the user, since you're not asking everyone to join the same group, and see each other's content. Only yours, and in turn theirs.
Matrix is basically a group chat with bells and whistles, which is really nice, but isn't what I'm looking for.
This service seems very fully featured, and I can't quite tell from reading if it does support what I'm looking for, so I'll just have to give it a try!
Thanks for sharing it :)