It’s actually not exactly true. Soap doesn’t break down oil. It attaches to the oil molecules, and attaches to a water molecule by the other end. Which, when the water is running away and takes all this mess into the drain, is incredibly effective. With the stagnant pool of water, less so.
I did wash the dishes in buckets when I was young, lived in poverty, and had to do it all by hands. I still remember that feeling of always dirty dishes, that’s why I am always terrified when people do it on purpose.
Reality unfortunately rarely follows “basic common sence”. And all the research shows that for most people in most cases restrictive diets only give temporary results.