I’ve been experimenting with Discord for a “virtual coworking” group. It’s nice because it’s always up, has message history, and has a lot of ways to access it. I like how I can control the invites and have an open forum too, which is something that Slack doesn’t allow without workarounds still to my knowledge. The main hurdle is explaining to people what Discord is (it doesn’t sound nice or like a group chat) and getting people through its weird invitation system because they keep reusing the invitation code on multiple devices. Slack might be a bit better to give the semblance of a closed network, especially if you are talking about kids.
I think though maybe the biggest challenge may be if the other couples like writing or chatting on their phones, otherwise they probably won’t feel compelled to connect. And also whether or not you want to put the energy into maintaining it; maybe it’s the job of the class-offering organization to point people toward a social solution.
Other thoughts: maybe Skype chat rooms would work; I used to use those to chat out-of-game, and they maintain history too. They may already have it working if they plan to share all kinds of baby-related video calls with people.