The specific combination of hotter water, rinse aid, and modern detergent mix (lipases, proteases, amylases, and surfactants) is actually able to remove certain chemicals that you couldn't otherwise remove via normal mechanical abrasion (at some point scrubbing harder with a harder abrasive will damage the dishes).
This is because the intermolecular bonds between certain food molecules (esp lipids like fats and oils) and the dish surfaces themselves can be remarkably strong. Additionally, it is quite possible for you to instantaneously remove molecules that are sticking due to intermolecular adhesive forces only for them to stick back on as you move said abrasive out of the way slightly.
The only real way you can actually remove such substances without mechanically damaging the dishes is to break down and/or dissolve them into the water so they don't restick to the surface.
This is because the intermolecular bonds between certain food molecules (esp lipids like fats and oils) and the dish surfaces themselves can be remarkably strong. Additionally, it is quite possible for you to instantaneously remove molecules that are sticking due to intermolecular adhesive forces only for them to stick back on as you move said abrasive out of the way slightly.
The only real way you can actually remove such substances without mechanically damaging the dishes is to break down and/or dissolve them into the water so they don't restick to the surface.