Group chats work well for a fixed set of users looking for all-to-all broadcast, but that's hardly a social network.

Something like Discord gets even closer. I'd say Discord actually makes a pretty good social network, although most people don't treat it as such. In my experience, the biggest issue with discord-as-a-social-network is the binary nature of discord server membership. I've had my one "friends group" server split into 3 or 4 servers (each with 80% membership overlap) over some people who had drama, and this makes it much less manageable as a social network because you tend to spend most of your time on one or maybe two servers.