Because dishwashers only use two "fills" of a few liters of water per cycle. Running your tap full-blast for 5 minutes will likely waste more water. Commercial grade dish basins are a bit better if you use them right, but still take more water to fill than a dishwasher.

Also a properly functioning dishwasher shouldn't require a hand-rinse if it's working correctly. Yeah, old & worn units can suck sometimes, but that's usually a problem with calcified pipes cutting the pressure to the unit, or you're not putting detergent in both of the little cups.