Ime restaurant napkins are not quite as absorbent as a cloth can be so I suspect some wax or lanolin admixture in the wash that directs the schmutz to the voids between the weave while armoring the fibres themselves. Also napkins are usually white so they can bleach.
Super effective and economical, although not safest degreaser, is gasoline. Turps for oil paints, obviously. Methylated alcohol for acrylic paint and window cleaning.
Also depends on the material. If you wear jeans then stop. Denim is the stupidest material ever invented. Change one tire and you'll have a slightly brown knee forever. All cotton basically sucks for any item of clothing.
Slow start but good discussion of water sheeting for paint on metal and piranha.