You also need to count the time for it to wash them and unload them. Washing is async which means its less active work, but it’s definitely includes in how long it takes.

The more dirty dishes you have the better the dishwasher becomes in terms of active work but in no scenario is it actually faster.

You do not need to count that time. There are very few situations where the time which is not active work time matters. Dishwasher uses up less of your time (and saves money, energy, water). Note the origin statement: It's just so much faster to do the washing by hand. It's completely wrong, but makes clear what kind of time is to be measured here.

> You also need to count the time for it to wash them

You do not. We are comparing the time it takes for a man to wash a dish versus the time it takes for that man to put that dish in the dishwasher, not versus the time it takes for a machine to wash that dish.