You have made reference to literally nothing which says or suggests that the three body sun-earth-moon problem can exist as a stable system.
Can you just clarify what you mean by a stable system? The Moon has been orbiting the Earth and the Earth orbiting the Sun for billions of years. Is that not evidence of a stable system?
You can see that the Moon is in orbit around the Earth just by observing it with your own eyes. As the Moon moves eastwards against the stars and the phase pattern matches the increasing and then decreasing angular elongation between the Sun and the Moon during the course of a month so natural and logical conclusion to come to is that the Moon is in orbit around the Earth.
It might be the case that the 3 body problem has never been solved on a piece of paper but it sure seems to be working in the sky. So long as we can predict exactly where the Moon and Sun are going to be at any one moment during any year then why is it significant if an unsolved 3 body problem an issue to you? You want evidence. The sky is your source of evidence. You can see what I and everyone else can see. If ancient astronomers could figure it out using nothing more than their own eyes then I'm sure we can give it a go as well.