The three-body problem cannot be solved exactly (unlike the two-body problem). 100 years of perturbations would result in statistical probabilities as meaningful as the weatherman predicting next September’s weather.
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.
bingo