Why could there not be two planets diametrically opposed in the same orbit?
Because, while your intuition may say otherwise, that's not a stable configuration. Even a small deviation will eventually blossom into one (or the other, or both) of the bodies falling out of such an arrangement.
"Clearing the neighborhood" is not an absolute thing - there are something like 10,000 near-Earth asteroids, for example. But, Pluto is orbiting around in the vicinity of many, many more Kuiper Belt objects. Its neighborhood isn't even remotely emptied.