I have nothing wrong with getting my emails faster, and I'm sure it would benefit a lot of people, but personally I don't want email to be realtime in the way that IMs or texting is. More importantly, I don't want anyone who emails me to expect it to be realtime.
When I want that kind of availability I have phone-availability and instant-message (be it phone-based or PC based). I'd rather keep email to important-yet-not-time-sensitive things.
If email were "realtime," I would have some kind of expectation coming from the people that want to use it in such a way, thereby beguiling me into constantly letting it be checked (by it alerting me).
Even if I don't allow it to pop up on my device I would still have to explain to each new contact that email for me does not mean an instantaneous read and reply.
Some of my friends get cranky when I don't answer their IMs instantly, I think it would be a bit of a pain if people treated email with the same expectations of responsiveness.