So the computer based solution proposed by Newton and Turing is rejected because the system is chaotic, meaning a general solution will always diverge from reality.
What I don't get is: this should still be good enough Run your solution every month or so, with updated measurements, and you'll have an ongoing "forecast" of conditions.
I'm referencing weather because that's we do. A weather forecast is a prediction of a chaotic system, but of one which changes every day or so. Prediction difficulty is dependent on local conditions and weather type, but we can still make predictions.
A gravitational system of 4 mutually interacting bodies is muuuuuch simpler than weather, and could be predicted far enough in advance to let a civilization adapt and persist!
Book spoilers.
That is not the reason why they had to leave their home planet. In fact, their calculations could go pretty far, but it also predicted that the planet will eventually become engulfed by one of the stars.
They further corroborated this with the observations that there used to be multiple planets in their star system (which have all since been engulfed by the stars) and the one that they are living on has been cracked into two by the last great cataclysm which further deteriorated whatever conditions it was on the original planet.