This happened to me last Friday morning driving to go turkey hunting. Ran straight into the side of my car by passenger mirror. I would have stopped and tried to get a tag if the deer went down. It ran away after causing $6,000 damage to the side of my car. I just wish you didn't have to get approval for a tag if you hit a deer with vehicle. I also don't have a problem if someone hits a deer and it lays. Wildlife usually cleans things up pretty well. Mostly eagles around here. Then again. We have idiots that would try to hit deer if they could keep them. You are right about the state claiming ownership of anything and everything most of the time though.
You'd have a hard time charging the person that hit the deer to remove it. I've hit several over the years and just kept driving with little or no damage to vehicle. The state and county highway department pick up everything around here, so with that being a state job we basically are paying for removing roadkill.