If you ray trace and shoot one ray in the center of the pixel then it will hit the white object, and no matter how many rays you spawn from that intersection point the part of the black object that's covering part of the pixel won't actually contribute to the final image. So with this technique the final pixel will be white.
With path tracing you shoot many initial rays at random sub-pixel positions, most miss the black object and hit the white one, but some hit the black one instead. So with this technique the final pixel will be gray, a mixture of the two objects' colors and therefore anti-aliased.
Since it's not much more computationally expensive to do sub-pixel sampling than to repeatedly sample the same point within the pixel you might as well just use sub-pixel sampling.
Additionally a pixel has no knowledge of its contents until you cast a ray. And even if the ray did hit a diffuse surface you would have to do more math to determine if the edge of that surface lies within the boundaries of that pixel. Might as well just use sub-pixel sampling.
In reality, pixels on the screen and camera sensor have an area. If you choose a random position on that area, you get anti-aliasing "for free" because there's going to be variation in the direction of rays that contribute to the same pixel.