It's got nothing to do with perturbations from additional small bodies, the problem is exactly mathematically calculating the outcome for exactly three bodies w

It's got nothing to do with perturbations from additional small bodies, the problem is exactly mathematically calculating the outcome for exactly three bodies within reasonable time frames even if you assume perfect ideal knowledge about the system. It turns out this is excruciatingly hard. The n-body problem where you add further bodies, even very small ones, is even harder.