No formula is not the same as no forecast: chaos, not the missing closed form, limits three-body prediction.

The claim

The three-body problem having no general closed-form solution does not make three-body motion unpredictable: the motion is deterministic and can be computed step by step. What limits prediction is chaos, the fast growth of small errors in the starting state, which sets a finite forecast horizon that depends on the configuration and on how precisely the state is measured.