Well, it both is and it isn’t hard.
It’s hard because the 3 body problem is impossible to solve for the general case. There are solutions for specific cases, i.e. identical planets in identical planets, but no exact solutions for Hohmann transfers in the real solar system. The issue is that every object in the solar system pulls on your spacecraft, and the result is so complex that it’s essentially chaotic.
It’s not so hard because most of those pulls are very small, and make very small changes to the course of the spacecraft. Your spacecraft can make a few small correction burns, and you can overcome all this tiny little influences from remote objects. It’s like driving - we’re all capable of adjusting the course of a car around a complex curve, because we’re making tiny little adjustments to the wheel in response to visual feedback. We’re not doing any internal math, it’s just a matter of moving the wheel until the car remains centered, and we do it so quickly we don’t even notice we’re doing it.
For the most part, you can do this remotely. It’s better and safer if the spacecraft computes it, but the 49 minute command delay isn’t important because you don’t have to respond in real time. Unless the spacecraft takes a significant impact from a micrometeorite, something big enough to transfer momentum, you can measure the spacecraft’s position and vector accurately enough remotely to compute correctional burns days before you need them.
The hard part of self-driving cars, the basic bits I mean, is that recognizing “this is a road” and “this is a lane” and “that is another car” are computationally difficult problems. Image recognition isn’t easy. By contrast, it’s really quite easy to write software to recognize a planet’s disc. It’s just the bright circle in the middle of your image. I’ve got telescope autoguiding software that does essentially that, only with stars, and it’s freeware that’s about 12mb. Which is gigantic by Apollo era standards (the Apollo flight computer had 72kb of ROM, and 4kb of RAM), but not by today’s.