Well ... that's why we need a mirror, but why do we need the shutter? If we are using a film camera then it's obvious, the shutter controls the length of time the light exposes the film for. With a sensor could it not simply be turned on, or sampled, for the length of time we need to make the correct exposure once the mirror is out of the way?
I'm all for speculation and without really knowing for sure that's all anyone here can do. So here are a couple of my guesses, from most to least likely;
1) It's comes down to money. The design of mechanical shutters has pretty much been perfected and the factories that build them are already in place and can pump them out relatively cheaply. The engineering and design of sensors and the electronics needed to replace shutters would not come cheap, so why bother.
2) Digital is so new they haven't had the time to perfect it yet.
3) It can't be done.
4) They haven't thought of it yet and I'm the first one to come up with the idea. Thank you, thank you very much.
5) Everyone loves a shutter. Who in their right mind would buy a camera without one? I like been a shutterbug, but a sensorbug? And what would happen to places like Lens and Shutter? Lens and Sensor! Not likely.
6) Kodak has a vested interested in shutters and just won't let go.
7) How else would we keep the dust (and our fingers) off our sensors? The temptation would be too great, seeing it peaking out from behind the mirror every time we changed a lens! Poke, poke. Yep, looks good to go. Pokie, poke.

The cultural changes required would be too great as we would no longer be able to say "shutter speed" and we would get confused.
Mitch