Your question inspired me to write a whole huge Topic about the experiment I’m in the process of with Crowdcast, hah! Thanks! You can read that here.
To answer you specifically:
I’ve actually tried to move away from facebook groups for several reasons. BUT, when I was focusing on bringing patrons into the group, I set it to be a private group but not secret (I think that’s the right terminology, it’s been a while), so that I could share the group link on the ‘new patron welcome’ page. They could then request membership and a moderator or I would let them in.
Regarding livestreaming, I think it is useful for both public and private streams. Public streams garner interest and grow your community. Private streams are SO fun cos they’re my patrons so they’re officially the loveliest people in my world! They’re fun, community-building, heart-warming and all the good things. They feel like a reward to myself as well as my patrons 
. Can you tell I like them? 
Anyways, hope that helps. Go read the full post, I think you might find some value in there, even if it doesn’t totally apply to where you’re at right now!
All the best to you 
Nate