In the case of the solar system, yes, it helps that the Sun is much more massive than everything else (and then Jupiter is 4 times more massive than Saturn, the next biggest) - you can go a long way to a "reasonable" solution by starting with the 2-body solution if only the Sun affected each planet, and then adding in the perturbation caused by Jupiter and Saturn. (In fact, that's how we predicted the existence of Neptune, by noticing that there were extra perturbations on Uranus beyond those, and hence another massive planet must exist, far enough away from the sun to only significantly affect Uranus).
Note that n-body problems are not particularly complex or hard to solve compared to other chaotic systems. In many ways, the existence or nonexistence of closed-form solutions is mostly a distraction: it merely reflects our choice of primitive functions and there is no sets of primitive functions that is stable for addition, multiplication, composition, inversion and integration. Typically, even the simple integral ∫ eˣ/x dx cannot be decomposed into more elementary functions.
But that doesn't matter in practice, because we are already using numerical approximation to compute primitive functions that are not implemented in hardware. Using numerical solvers to compute solutions to ODE is not so different. A good illustration of that point is that there is an analytic solution for the 3-body problem (in the form of an infinite series in t^{1/3}). But this solution is useless for computing orbits because it has bad convergence properties. In other words, it is better to use a numerical solver rather than stick to the analytical solution. And a similar phenomenon exists for polynomial equation of degrees 3 and 4: the exact formula is numerically unstable, and its better to use a numerical solver when one wants a numerical solution.
One such case where a solution is known is the Lagrange point of the Earth-Moon-Sun system (and similiarly for other points) https://en.wikipedia.org/wiki/Lagrange_point But in reality, they exist only as an approximation. They aren't truly stable.
My understanding the way to calculate spacecraft, asteroid, etc. trajectories is just through a discrete simulation.
Like f you don't know how to solve the antiderivative of a given function, you can still calculate the integral since you know the value of the function.
Computers can easily solve initial value problems for most ordinary differential equations. They integrate them, calculating an approximate solution after every small, but finite step.
Getting an approximate solution to the 3-Body problem can be achieved in around 20 lines of python, without having to use any libraries. It is a remarkable simple and effective technique.