I suppose it would be like a golf club that bends if you get out of the groove. I don't know cricket, but if you hit a ball with a baseball bat too high on the ball, the ball goes into the ground, too low on the ball, it pops up. If you swing too early, you foul to the left, too late and you foul to the right. Too me that's the feedback you need to get better at batting and using a regulation bat is the best way to get that feedback. Different bats have different sweet spots, and I suppose using one with a smaller sweetspot than the one you use in a game is beneficial. I suppose different coaches could have different ways of doing things. Whatever works.
I think the narrow bat in practice forces one to hit with the center of the bat. Cricket bats are almost flat, with a slight curve. I've heard of baseball players practicing with a dowel, which is, I suppose a similar idea. Too much time spend with something a lot lighter than regulation would throw off the swing.