If the 3-Body force problem hasn't been solved, how do rocket scientists plan orbits of spacecraft?

The three body problem isn’t “solved” in the sense that there is no known closed form solution that works for any general initial conditions.

However, when you have two massive bodies and one that is considerably lighter, you can estimate the trajectory with almost any degree of accuracy. Furthermore, numerical techniques will allow you to do that with multiple bodies and without any restrictions on their mass.

The analysis roughly comes down to this:

  • at a given instant the position and velocity of all objects is known (the initial conditions)
  • from the positions we know the gravitational forces on each object (magnitude and direction)
  • given these forces and the mass of each object, we can compute the acceleration at this instant
  • from the position, velocity and instantaneous acceleration, we can compute the velocity and position a very short time later
  • repeat for small time steps to compute the complete orbit (obviously, if you are firing the rocket you also need to take account of the thrust and the changing mass)

There is an entire field of scientific computing dedicated to doing this right. And the method briefly made an appearance in the movie “Hidden figures” - at that time these things were still under development, and of course computers were much slower, and had less memory, than today’s machines.

I have written several answers on this site that used such an approach. Probably the most relevant is this one

What methods would they use to predict what would happen in a situation when a probe is being acted upon by the gravity of two stars, say?

For that matter, what methods do they use to predict what would happen in a situation when a probe is being acted upon by the gravity of a single star plus eight planets, a large number of moons, and a very large number of asteroids and comets?

That is a rhetorical question. It's a bit false to say that the N body problem cannot be solved. It does have a series solution, developed first for the three body problem over a century ago, and generalized to N bodies over a quarter of a century ago, but nobody uses it. Too many terms are needed.

Even without that series solution, there are ways to solve the N body problem in practice, at least for a short period of time. In the case of our solar system, that short period of time is a few million years. (That qualifies as "short" compared to the age of the solar system.)

In a way, this is no different than almost every interesting problem in physics. Problems of interest that have closed form solutions are few and far between. For everything else, some kind of approximation technique is needed.

A large number of approximation techniques exist in the case of orbital mechanics. The approach outlined in Floris' answer is but one. Much more accurate but significantly more complex techniques are used to model probes sent from the Earth to other planets.

That said, any technique used to predict where planets and probes will be in the future will inevitably be imperfect. Compounding the problem, models of the thrust generated by a spacecraft inevitably are imperfect as well, as are instruments that measure that thrust. Any approach used to guide a probe to another planet must necessarily deal with those imperfections. Techniques that address these issues also exist. The development of these techniques was one of the key challenges in getting men to the Moon in the 1960s.