What exactly do you mean the driver is responsible for hitting something? And not stopping your car when you should have, what does that mean? So if a driver is operating his vehicle legally (driving where he should be, going legal speeds, etc.) and someone or something enters the roadway illegally (e.g. a jaywalking person or dog off leash) the driver is expected to stop his car immediately (never mind the laws of physics) or he/she is responsible? I'll defer to any expert insurer's opinion, but I'm pretty sure that once an insurance company pays the driver to cover his damages, that they would then attempt to recover damages from the person found to be legally responsible.
There are obviously lots of what ifs and omissions of detail in the reported story that prevent any firm conclusion about justice in this situation. It is interesting that where some people view this a "searching for blame" issue, others (like me) see it as an "avoiding responsibility" issue.
Stop yelling at me.
Two incidents come to mind, and both happened to coworkers, not me, so I could be wrong.
1. Coworker's husband is driving on a country-ish road on his way to work in early morning. A deers chooses that precise moment to run across the road, and hits this man's car. He didn't hit it; it hit him, damaging the front quarter panel. Coworker told me that the insurer wanted to know, in clear terms, who hit what, and how. Apparently, the coverage (and increase in premiums) would have been different had the driver done the hitting.
2. Coworker driving in heavy traffic. Car ahead comes to a dead stop; he can't stop in time and hits it, but at low speed. Insurer tells him the onus was on him to maneuver his own car properly, regardless of the sudden stop ahead of him. I tend to agree.
Again, I could be wrong. I will certainly defer to the experts as well.