Yeah my argument is that this is nonsense. Just turn the tap off often when you’re scrubbing the dishes (use the water for rinsing) and it should be obvious that you’re using less water — particularly when you take into account the fact that a dishwasher uses 4 gallons of water no matter how many dishes are in it.
Under ideal conditions perhaps a modern dishwasher comes out ahead, but again maybe the ideal Rambo bird-killing-machine cat is also capable of decimating local bird populations.