It takes me more than 3 seconds to clean and dry my mug of tea and my spoon and they are not exactly dirty.
I admit I'm guessing the seconds.
Actually a tea-cup does take me longer than a dinner plate, as tea seems to stain the cup and I need to rub longer with a scourer to get it clean.
Let's say it takes 10 seconds per item then.
Plate
Knife
fork
mug
teaspoon
pot 1
pot 2
That's 70 seconds for one person, 50 seconds for each additional person, so hypothetical family of four that's still under four minutes. (I'm not including drying and putting away because I don't do either.)
OK I've left some things out, there will be knives/spoons etc used in preparation and serving. We're not talking whole extra minutes omitted though.
If washing by hand took 20 minutes (as someone up-thread said and as I remember it taking when done the traditional way in a sink, followed by drying and putting away) then I'd agree a washer might be better.
I think how we cook probably helps us. We are doing one-pot dishes, usually Asian. My mother did traditional English cooking and that seems to involve far more pots. (We use an oven maybe once a month. Cleaning ovenware does take a bit more effort.)