does this imply that most solar system arrangements are unstable and we are lucky in ours?
No, not really.
Even without knowing the masses of the planets and the sun here, the scale of this game is tiny compared to the real solar system. In realistic scales, it is a lot easier to come up with stable configurations.
Planetary systems form slowly from a gas and dust disk surrounding a new born star. The early protoplanets will collide with each other and form bigger bodies and the system will slowly reach a stable configuration. Planets may be thrown out of the system or flung into the star too.
The best current understanding of exoplanet systems is that there are planetary systems orbiting the majority of stars out there. Our solar system is neither an extremely lucky coincidence or a rarity among other stars.
All the exoplanet systems were discovered only recently (span of a few decades). So we don't really know if those systems are stable in the long run, and hence we don't really know if our solar system is "lucky" or not.
Yes, all exoplanet systems have been discovered recently but the best theories explaining the birth of planetary systems predict that the planets form very soon after the formation of the central star. And we have good estimates about ages of stars, which are several orders of magnitude (as in 10^8 vs. 10) older than observations about exoplanet systems.
Statistically that means that we were not "lucky" and neither were any of the other exoplanet systems.
Given that the science behind exploring exoplanet systems is so young, this theory is not necessarily correct but it is the best understanding that the scientific community currently has.
We are lucky that our stable arrangement happens to have a planet with surviving water within the habitable zone and with a few elements in the system that make collisions less likely. (Both the Jupiter and the Moon "sweep" space so that collisions with earth are much less common than they would be without them.)
Of course, the anthropic principle is in full force. The earth had to be lucky for you to stand on it wondering about how lucky it was -- so, if it hadn't been lucky, you would not do so and maybe some green-skinned alien halfway accross the galaxy would spend time wondering how lucky his planet was.
Come and get me, copper.
Exactly. For instance, there was a time when Earth didn't have a Moon, which would make collisions more likely. At least one of these collisions seems to have created the Moon. We of course are familiar with the version of Earth that has a Moon; it is unlikely that we would have evolved without that Moon such that we were left thinking "Gee, it would be nice to have a moon.."
I also think that the solar system isn't entirely stable. The orbits of the planets do influence one another in a way that isn't in equilibrium. It just happens very slowly.