Just add some LDAP services for identity management, authentication and it's fairly close to the project I'm on right now. A cursory look at getting a matrix server running didn't work out yet though, with issues for clients to recognise the server at all. I don't mind sysops work but right now a set&forget solution would spare some needed energy.

The Thunderbird+Matrix integration sounds like a match - I hope that holds up in practice as well!

Edit: as others mentioned, Nextcloud has the features.