I've thought this too. Regarding weight distribution specifically, I've never found the front wheel to be in a position where it has superior traction over the rear wheel. Yes, I've had a rear wheel spin out on me on a mountain bike before, but as soon as I change the weight distribution the problem is fixed. Also, any situation I can think of where "AWD" would be "handy," you'd likely be moving incredibly slow, meaning you are going uphill, in low gear, pedaling quickly. I would think having the front wheel driving would be more of a hindrance here, gripping at random rocks and sticks and potentially making the ride more unstable and dangerous.
Really what I'm saying is that riding a bike is a dynamic activity and it seems that only providing power to one of two wheels is the "right answer."