Especially for New Horizons I would say Jupiter gravity sssist was one of "safest" stages of the mission. :) Because New Horizons' closest flyby of Jupiter was rather far - at 2.3 million kilometers.
Gravity assists require specific trajectory of flyby - with specific direction and velocity. How precize should they be, how large is error margin - it depends on how close is flyby to the "assisting" planet. If it's very close - than precision is critical, because gravity potential changes steeply there, and small error in pre-assist trajectory will result in great changes of post-assist trajectory. The changes can be so great that probably the spacecraft won't be able to correct the trajectory by propulsion. Also error in close flyby can result in the spacecraft impacting the planet and being destroyed. But if flyby is far enough - than gravitation changes slowly with distance to the planet and so error margin in spacecraft trajectory is much bigger.
Trajectory for gravity assists is usually ajusted gradually. There are planned 2-3 or more trajectory correction maneuvers (TCM) for every specific gravity assist. For example first corection several months before flyby, second couple of weeks before and third couple of days before. The last maneuvers are usually very small and sometimes even being canceled because the trajectory is already fine.
So for New Horizons - error margin for Jupiter gravity assist would be large enough. If we imagine NH would fall out of communication for several weeks during Jupiter flyby and could not perforn the final trajectory corrections - I expect trajectory error would not be so big and NH could correct it later by propulsion and reach Pluto.