The moon’s orbit is taking it further from Earth over time, it it’s very far away and very massive, as well as being influenced by the Sun. A satellite’s mass is almost negligible, and compared to the Earth-Moon distance it’s distance from Earth is too. A pebble hitting the moon also isn’t going to do much to change its course, while the same pebble will do a lot more to a satellite. The same is true of dust.

The moon has a ton of mass and—therefore—inertia. The chaotic effects of the system are still present, but require dramatically larger timescales to notice any kind of effect.

The moon's stability is a good point, but not conclusive. The moon's orbit isn't perturbed by it's own gravity. It's also much larger, so much more mass (radius cubed)/surface area (radius squared), so less affected by solar wind, micro meteor impacts, and the like.

But still, it's more likely we'd find evidence of a distant past civilisation on the moon than on earth right?