I didn't program the forums. But that's something Rev might be able to look at with all the work he is doing on them, I don't know.
I'll look into doing that, though I'm hoping there'll be an existing mod that extends the messaging system.
Implementing that is likely to involve changing some of the core forum functions rather than just the display templates, which while not necessarily dangerous (I'd develop it on my test forum), can create a lot of work ensuring compatibility when upgrading to new versions.
I manage a Zen Cart shop and a couple of other forums (PhpBB), and testing that the extensions I've written with the new versions is always tedious. Maybe just making it save the messages to the Outbox by default would be sufficient?