When a divergence is identified the dynamic trajectory expansion terminates and a new state is chosen from the trajectory that has been build up to that point – it does not just take the initial point.
But more importantly as @sakrejda notes, a divergence is not a point but a property of the trajectory itself. There is no discrete threshold below which the trajectory is fine and above which it’s bad – it’s the entire trajectory (really the entire modified energy level set) that’s pathological. We define an arbitrary threshold tuned to catch these divergent trajectories but the location where the trajectory passes that threshold is not necessarily indicative of the neighborhood where the problem first arose.
This is why the recommendation is to focus on where the divergent trajectories concentrate, for which our best proxy is observing the samples that came from divergent trajectories.
Trying to extract more information about the divergent trajectories is straightforward but there’s no place to put that information. In particular, you cannot replace the carefully-chosen sample with an arbitrary point from the divergent trajectory as that will immediately break detailed balance, and we don’t have any addition sources where dynamic information like this could be dumped.