The Blackberry managed to be a unified inbox because it also unified UX.

Today everyone has to have their own UX, so the notion of, say, message _transports_ (which is what you had in, for instance, cross-protocol IM clients like Adium) is dead.

iMessage, Facebook, Whatsapp, Signal, etc. all have their own UX, and that is impossible to unify today.

Android _almost_ got that right as you could (theoretically) build unified front-ends to front for separate activities back in the early revisions (and I believe PalmOS nearly got there as well) but today every service wants to own the UX to shove their "experience" in your phone...