Sorry about your unfortunate incident, but I guess I can understand why the law might look on this as just an accident and not something for which the owner of the animal would be criminally liable. No requirements out in the country to leash or control animals, so obviously the law looks at it as if you had hit a deer or something similar. I don't know that I totally agree with that thought process, but it is what it is. Just be thankful the result was repairable damage to the bike, and not major damage to the rider.
I've always been very careful when riding out in the unincorporated areas around here just for the reason you mentioned. People who live in those areas tend to be used to doing as they please and also tend to not think about the fact that someone may be driving by at any time. I've had to dodge dogs, kids on scooters or ATV's shooting out of driveways, people pulling out of driveways in cars and trucks with nary a glance to see if there is any traffic, etc. While riding in the country is much more enjoyable than riding in towns and cities, it can be just as dangerous.
Hopefully your insurance is good and they'll get you fixed up ASAP.
Jerry