Example: Why individual outcomes are insufficient

On a one-dimensional input space, take the two four-outcome POVMs

Their total-variation distance is $1$, whereas the largest individual-outcome discrepancy is $1/2$. The event $\{1,2\}$ attains the correct value. Thus ${\max }_w\Vert D_w{\Vert }_{\infty }$ is not the general formula. For two outcomes, it does suffice because $D_1=-D_0$.