When I use a regular email account I practice inbox zero, and delete messages. I never store information in an inbox; rather I moved it to text files or a wiki. Since Discourse is all the parts of a wiki I use, and a whole lot more on the messaging side, I just skip a step.
On top of using it for incoming messages (a bit like a CRM), I also load up all my peeps on a personal but public instance. They get free discussion hosting, and I get to send PMs to folks by their username, even from my phone, and if they ever change their email address it auto-updates! Wow!
Notably, I don’t allow Discourse to ever send email to me; it would just go to another Discourse.
Just an inbox run by a registrar. I think they use roundcube or another web interface, but I don’t use it.
Kinda. I’ve experimented with webhooks to ping me on jabber for certain criteria, like special clients or keywords. But I honestly haven’t needed that. I do use auto-bumping and topic timers, and I split different addresses to categories that have slightly different settings (including not notifying me).
I also post things, via webhooks, from various systems, so I can see it all in one place.
My favorite functionality shift is the ability to “close” conversations. I also like silencing spammy addresses.