Legal or not, I still think it's kinda cruel, and I personally couldn't do it. Legal and right are not always on the same side. Just what IF dogs were not considered a "property". Some groups would love that. And it's not unthinkable. And it would only mean a change on the legal issue, not the moral one.
But the OP is not "cruel" for wanting to be made whole for an accident that was not his fault and could have been prevented by reasonable measures on the part of the pet owner.
Is there some sort of a time limit for that responsibility? Like if the dog ran away and couldn't be found for a month and then gets "found" (killed) by the car? Or does there have to be a clear connection between the events?