You do an accounting of the nearby massive objects and their distances from each other, pairs of objects within certain bounds of mass and distance can be treated like there are only those two objects in the universe with the understanding that there will be a small error because of outside influences. It’s an approximation, but in the right circumstances a very good one.

Each planet and the sun can be done like this, ignoring all of the other planets. Each moon and its planet can be considered a two body system ignoring the rest of the moons.

If you just randomly generated a bunch of massive bodies and pressed play, you would have few 2 body systems and a lot of chaos, but that’s a problem that solves itself as the chaos results in either collisions or ejections.

You're taking it too literally. The 'nudge' means a change in initial conditions, and could result e.g. from measurement uncertainty, not necessarily a literal nudge.

It also doesn't necessarily matter that much if there are more than two bodies, if the gravitational influence of other bodies is small enough, then you can model as a two body problem.

The “nudge” can also come up even with exact starting information. When you calculate the next step and round it to the nearest nanometer, you’ve just nudged the system by enough to eventually make your prediction worthless in the 3-body case.

The relative magnitude of interactions is important. The article specifically mentioned triple star system as an example. Solar system is of course many body, but (a) it has already settled into (sort of) stable configuration (b) there is one massive body and everything else revolves around it. Yes, massive bodies like gas giants have noticeable affect on other bodies, but due to the distances they tend to be in the stable territory.