If I post a picture of my dog on Facebook and my child's teacher sees, that feels normal since they just kind of happen upon it. If I send them an email with a picture of my dog, that feels like I'm over stepping a bit.
So how about a pull model for email: newsletters. I'm imagining a service like Substack for regular folks. People could sign up for your personal/family email newsletter. This is like a mailing list, but point to point. Recipients could reply to you, the sender, but not "Reply All" to your full network.
I like it but forget the whole "network" part. Double opt-in only with the owner of the account not able to upload or add email addresses. That might stop it for a little while from becoming spam and have people that truly want to hear from you.
"Double opt-in" meaning a potential recipient would submit a sign-up request and then you, the author, would approve or deny their request? That would be good for privacy and reducing spam, but not allowing the author to upload their own set of email addresses would make bootstrapping your early list harder. Or perhaps the author can upload email addresses, but the service sends an email to potential recipients, asking them to confirm they want to sign up.
To bootstrap your list and make the service more viral, the service could cross-post the newsletters (or just an excerpt) to Facebook (and other social media services) with a newsletter sign-up link telling people they could receive your full updates in email without having to log into Facebook.
It's hardly easy with email. Keeping an up to date list of working addresses becomes a huge hassle once you get beyond a few people. I remember trying that in the days before Facebook and every time I sent a message it would bounce for some recipients.
Because people are already posting to Facebook, and it provides a lot of excellent scaffolding, including a small comment box that gives friction to large messages, and little friction to short messages. A bunch of short messages is what I want, which is not something typically someone is going to email. They also want to broadcast it to all their friends in a lightweight way that doesn't clog up inboxes and signals that it's optional knowledge, which an email doesn't.
Easy to do with email. Why do you need Facebook at all if those are your objectives?