A three body problem cannot be solved “analytically”, but numerical solutions are possible to arbitrary precision. There are a few models of the solar system’s gravity available, and NASA may have the second best, but uncertainties are pretty well characterized. Except in the cases of a close call, I don’t think that accuracy is necessarily the issue, it’s just avoiding publishing any catastrophic predictions centuries into the future with a government imprimatur. (Not that global warming alarmists feel any constraints.) If there is an issue there are lots of academics who can provide the appropriate level of sensationalization.
The issue with the N-body (N > 2) problem is that it is sensitive to initial conditions. In other words, a chaotic system.
No matter how accurate your approximate solutions are, if you don’t have highly accurate initial conditions, you can have wildly varying results in a relatively short period of time.