The explanation is that there are two desired attributes at work: yield stress and shear rate dependency. These physical properties account for several aspects of flow behavior. A sufficiently high yield stress allows the pigments, resins and other additives to remain suspended in the paint and stay on the artist's brush after dipping. The stationary tongue depressor in the paint jar confirms this. A shear thinning paint allows the paint to transfer easily onto the canvas. Artists would complain if this were not the case.
Consider what an artist would ideally want the paint to do: be easily mixed by the user (simple shaking or gentle stirring), readily attach itself to the brush when dipped into the paint, flow without spreading beyond the width of the brush when applied and stop flowing without dripping when the brush is lifted off the canvas. These actions are a combination of imposed shear stress and shear rate. If the yield stress of the paint is too high, the paintbrush may splay or go into the fluid at an angle other than along the path of insertion; if the yield stress is too low, then the paint may fall off the brush after removal.