The abusive father should definitely be responsible for being abusive and if that abuse breaks some type of law, then they should be punished for their actions. But should the father be responsible for every bad thing their kid does as an adult?

If adults who were abused as children become alcoholics, and then they drove drunk and killed someone, it's the kid who should be arrested and not the father.

That's not to say that we shouldn't have support systems in place for people to go get help, but moral responsibility has a clear line to me.

By your logic, a culture that teaches people they aren't responsible for their own actions would bear responsibility as well.

You are presenting a false dichotomy. There is lots of room in between "no responsibility" and "sole responsibility".