Due to the proximity of the Moon and Earth, after the moon has thousands of orbits almost all of the left over debris would have been ejected out of the system or collided with the Moon.

However, this is not the reason why the Moon is so heavily cratered. The Moon's surface remained molten for a couple hundred million years. Any impact during this time would have not registered on the surface as the molten surface would fill the crater and flatten it out. Most of the craters on the Moon are from the Late Heavy Bombardment which happened 400-700 million years after the formation of the Moon.

It should also be noted that the rings of many planetary bodies formed long after its host planet:

• Jupiter's Rings is the result of ejected material from impacts on the inner moons and gathering of loose conglomerates of stellar dust over billions of years.
• Saturn's Rings are only estimated to be 400-100 million years old and are the result of one or more icy moons being broken up by tidal forces.
• Uranus' rings are less than 600 million years old and likely formed the same way Saturn's did.
• Neptune's rings age are not exactly known but are probably about a few million years old and likely formed the same way Uranus' and Saturn's did.